# Big Ideas Math Geometry Answers Chapter 1 Basics of Geometry

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## Big Ideas Math Book Geometry Answer Key Chapter 1 Basics of Geometry

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### Basics of Geometry Maintaining Mathematical Proficiency

Simplify the expression.

Question 1.
|8 – 12|

Question 2.
|- 6 – 5|

Question 3.
|4 + (-9)|

Question 4.
|13 + (-4)|

Question 5.
|6 – (- 2)|

Question 6.
|5 – (- 1)|

Question 7.
|- 8 – (- 7)|

Question 8.
|8 – 13|

Question 9.
|- 14 – 3|

Find the area of the triangle.

Question 10.

Question 11.

Question 12.

Question 13.
ABSTRACT REASONING
Describe the possible values for x and y when |x – y| > 0. What does it mean when |x – y| = 0 ? Can |x – y| < 0? Explain your reasoning.

### Basics of Geometry Mathematical Practices

Monitoring Progress

Question 1.
Find the area of the polygon using the specified units. Round your answer to the nearest hundredth.

Question 2.
parallelogram (square centimeters)

Question 3.
The distance between two cities is 120 miles. What is the distance in kilometers? Round your answer to the nearest whole number.

### 1.1 Points, Lines, and Planes

Exploration 1

Using Dynamic Geometry Software

Work with a partner: Use dynamic geometry software to draw several points. Also draw some lines, line segments, and rays. What is the difference between a line, a line segment, and a ray?
Sample

Exploration 2

Intersections of Lines and Planes

Work with a partner:

a. Describe and sketch the ways in which two lines can intersect or not intersect. Give examples of each using the lines formed by the walls. floor. and ceiling in your classroom.

b. Describe and sketch the ways in which a line and a plane can intersect or not intersect. Give examples of each using the walls. floor, and ceiling in your classroom.

c. Describe and sketch the ways in which two planes can intersect or not intersect. Give examples of each using the walls. floor, and ceiling in your classroom.

Exploration 3

Exploring Dynamic Geometry Software

UNDERSTANDING MATHEMATICAL TERMS
To be proficient in math, you need to understand definitions and previously established results. An appropriate tool, such as a software package, can sometimes help.

Work with a partner. Use dynamic geometry software to explore geometry. Use the software to find a term or concept that is unfamiliar to you. Then use the capabilities of the software to determine the meaning of the term or concept.

Question 4.
How can you use dynamic geometry software to visualize geometric concepts?

### Lesson 1.1 Points, Lines, and Planes

Monitoring Progress

Question 1.
Use the diagram in Example 1. Give two other names for . Name a point that is not coplanar with points Q. S, and T.

Question 2.
Give another name for $$\overline{K L}$$.

Question 3.
Are $$\vec{K}$$P and $$\vec{P}$$K the same ray? Are $$\vec{N}$$P and $$\vec{N}$$M the same ray? Explain.

Question 4.
Sketch two different lines that intersect a plane at the same point.

Question 5.
Name the intersection of  and line k.

Question 6.
Name the intersection of plane A and plane B.

Question 7.
Name the intersection of line k and plane A.

Monitoring Progress

Use the diagram that shows a molecule of phosphorus pentachloride.

Question 8.
Name two different planes that contain line s.

Question 9.
Name three different planes that contain point K.

Question 10.
Name two different planes that contain $$\vec{H}$$J.

### Exercise 1.1 Points, Lines, and Planes

Vocabulary and Core Concept Check

Question 1.
WRITING
Compare collinear points and coplanar points.

Question 2.
WHICH ONE DOESN’T BELONG?
Which term does not belong with the other three? Explain your reasoning.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6, use the diagram.

Question 3.
Name four points.

Question 4.
Name two lines.

Question 5.
Name the plane that contains points A, B, and C.

Question 6.
Name the plane that contains points A, D, and E.

In Exercises 7 – 10. use the diagram. (See Example 1.)

Question 7.
Give two other names for .

Question 8.
Give another name or plane V.

Question 9.
Name three points that are collinear. Then name a fourth point that is not collinear with these three points.

Question 10.
Name a point that is not coplanar with R, S, and T.

In Exercises 11 – 16, use the diagram.

Question 11.
What is another name for $$\overline{B D}$$?

Question 12.
What is another name for $$\overline{A C}$$?

Question 13.
What is another name for ray $$\vec{A}$$E?

Question 14.
Name all rays with endpoint E.

Question 15.
Name two pairs of opposite rays.

Question 16.
Name one pair of rays that are not opposite rays.

In Exercises 17 – 24, sketch the figure described.

Question 17.
plane P and line l intersecting at one point

Question 18.
plane K and line m intersecting at all points on line m

Question 19.

Question 20.
$$\vec{M}$$N and $$\vec{N}$$X

Question 21.
plane M and $$\vec{N}$$B intersecting at B

Question 22.
plane M and $$\vec{N}$$B intersecting at A

Question 23.
plane A and plane B not intersecting

Question 24.
plane C and plane D intersecting at

ERROR ANALYSIS
In Exercises 25 and 26, describe and correct the error in naming opposite rays in the diagram.

Question 25.

Question 26.

In Exercises 27 – 34. use the diagram.

Question 27.
Name a point that is collinear with points E and H.

Question 28.
Name a point that is collinear with points B and I.

Question 29.
Name a p0int that is not collinear with points E and H.

Question 30.
Name a point that is not collinear with points B and I.

Question 31.
Name a point that is coplanar with points D, A, and B.

Question 32.
Name a point that is coplanar with points C, G, and F.

Question 33.
Name the intersection of plane AEH and plane FBE.

Question 34.
Name the intersection of plane BGF and plane HDG.

In Exercises 35 – 38, name the geometric term modeled by the object.

Question 35.

Question 36.

Question 37.

Question 38.

In Exercises 39 – 44. use the diagram to name all the points that are not coplanar with the given points.

Question 39.
N, K, and L

Question 40.
P, Q, and N

Question 41.
P, Q, and R

Question 42.
R, K, and N

Question 43.
P, S, and K

Question 44.
Q, K, and L

Question 45.
CRITICAL THINKING
Given two points on a line and a third point not on the line. is it possible to draw
a plane that includes the line and the third point? Explain your reasoning.

Question 46.
CRITICAL THINKING
Is it possible for one point to be in two different planes? Explain your reasoning.

Question 47.
REASONING
Explain why a four-legged chair may rock from side to side even if the floor is level. Would a three-legged chair on the same level floor rock from side to side? Why or why not?

Question 48.
THOUGHT PROVOKING
You are designing the living room of an apartment. Counting the floor, walls, and ceiling, you Want the design to contain at least eight different planes. Draw a diagram of your design. Label each plane in your design.

Question 49.
LOOKING FOR STRUCTURE
Two coplanar intersecting lines will always intersect at one point. What is the greatest number 0f intersection Points that exist it you draw tour coplanar lines? Explain.

Question 50.
HOW DO YOU SEE IT?
You and your friend walk in opposite directions, forming opposite rays. You were originally on the comer of Apple Avenue and Cherry Court.

a. Name two possibilities of the road and direction you and your friend may have traveled.
b. Your friend claims he went north on Cherry Court. and you went east on Apple Avenue. Make an argument as to why you know this could not have happened.

MATHEMATICAL CONNECTIONS
In Exercises 51 – 54. graph the inequality on a number line. Tell whether the graph is a segment a ray or rays. a point, or a line.

Question 51.
x ≤ 3

Question 52.
– 7 ≤ x ≤ 4

Question 53.
x ≥ 5 or x ≤ – 2

Question 54.
|x| ≤ 0

Question 55.
MODELING WITH MATHEMATICS
Use the diagram.

a. Name two points that arc collinear with P.
b. Name two planes that contain J.
c. Name all the points that are in more than One plane.

CRITICAL THINKING
In Exercises 56 – 63. complete the
statement with always, sometimes or never. Explain your reasoning.

Question 56.
A line ________________ has endpoints.

Question 57.
A line and a point _________________ intersect

Question 58.
A plane and a point ________________ intersect.

Question 59.
Two plaes _________________ intersect in a line.

Question 60.
Two points ____________________ determine a line.

Question 61.
Any three points ____________________ determine a plane.

Question 62.
Any three points not on the same line ____________________ determine a plane.

Question 63.
Two lines that are not parallel _________________ intersect.

Question 64.
ABSTRACT REASONING
Is it possible for three planes to never intersect? intersect in one line? intersect in one point? Sketch the possible situations.

Maintaining Mathematical Proficiency
Find the absolute value.
Question 65.
|6 + 2|

Question 66.
|3 – 9|

Question 67.
|- 8 – 2|

Question 68.
|7 – 11|

Solve the equation

Question 69.
18 + x = 43

Question 70.
36 + x = 20

Question 71.
x – 15 = 7

Question 72.
x – 23 = 19

### 1.2 Measuring and Constructing Segments

Essential Question
How can you measure and construct a line segment?

Exploration 1

Measuring Line Segments Using Nonstandard Units

Work with a partner.

a. Draw a line segment that has a length of 6 inches.

b. Use a standard-sized paper clip to measure the length of the line segment.Explain how you measured the linesegment in “paper clips.”

c. Write conversion factors from paper clips o inches and vice versa.

d. A straightedge is a tool that you can use to draw a straight line. An example of a straightedge is a ruler. Use only a pencil, straightedge, paper clip, and paper to draw another line segment that is 6 inches long. Explain your process.

Exploration 2

Measuring Line Segments Using Nonstandard Units

Work with a partner.

a. Fold a 3-inch by 5-inch index card on one of its diagonals.

b. Use the Pythagorean Theorem to algebraically determine the length of the diagonal in inches. Use a ruler to check your answer.

c. Measure the length and width of the index card in paper clips.

d. Use the Pythagorean Theorem to algebraically determine the length of the diagonal in paper clips. Then check your answer by measuring the length of the diagonal inpaper clips. Does the Pythagorean Theorem work for any unit of measure? Justify your answer.

Exploration 3

Measuring Heights Using Nonstandard Units

Work with a partner.

Consider a unit of length that is equal to the length of the diagonal you found in Exploration 2. Call this length “1 diag.” How tall are you indiags? Explain how you obtained your answer.

Question 4.
How can you measure and construct a line segment?

### Lesson 1.2 Measuring and Constructing Segments

Monitoring Progress

Use a ruler to measure the length of the segment to the nearest $$\frac{1}{8}$$ inch.

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.
Plot A(- 2, 4), B(3, 4), C(0, 2), and D(0, – 2) in a coordinate plane. Then
determine whether $$\overline{A B}$$ and $$\overline{C D}$$ are congruent.

Question 6.
Use the Segment Addition Postulate to find XZ.

Question 7.
In the diagram. WY = 30. Can you use the Segment Addition Postulate to find the distance between points W and Z? Explain your reasoning.

Question 8.
Use the diagram at the left to find KL.

Question 9.
The cities shown on the map lie approximately in a straight line. Find the distance from Albuquerque. New Mexico. to Provo. Utah.

### Exercise 1.2 Measuring and Constructing Segments

Question 1.
WRITING
Explain how $$\overline{X Y}$$ and XY arc different.

Question 2.
DIFFERENT WORDS. SAME QUESTION
Which is different? Find “both” answers.

Find AC + CB
Find BC – AC
Find AB
Find CA + BC.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6, use a ruler to measure the length of the segment to the nearest tenth of a centimeter.

Question 3.

Question 4.

Question 5.

Question 6.

CONSTRUCTION
In Exercises 7 and 8. use a compass and straightedge to construct a copy of the segment.
Question 7.
Copy the segment in Exercise 3.

Question 8.
Copy the segment in Exercise 4.

In Exercises 9 – 14, plot the points in a coordinate plane. Then determine whether $$\overline{A B}$$ and $$\overline{C D}$$ are congruent.
Question 9.
A(- 4, 5), B(- 4, 8), C(2, – 3), D(2, 0)

Question 10.
A(6, – 1), B(1, – 1), C(2, – 3), D(4, – 3)

Question 11.
A(8, 3), B(- 1, 3), C(5, 10), D(5, 3)

Question 12.
A(6, – 8), B(6, 1), C(7, – 2), D(- 2, – 2)

Question 13.
A(- 5, 6), B(- 5, – 1), C(- 4, 3), D(3, 3)

Question 14.
A(10, – 4), B(3, – 4), C(- 1, 2), D(- 1, 5)

In Exercises 15 – 22. find FH.
Question 15.

Question 16.

Question 17.

Question 18.

Question 19.

Question 20.

Question 21.

Question 22.

ERROR ANALYSIS
In Exercises 23 and 24, describe and correct the error in finding the length of $$\overline{A B}$$.

Question 23.

Question 24.

Question 25.
ATTENDING TO PRECISION
The diagram shows an insect called a walking stick. Use the ruler to estimate the length of the abdomen and the length of the thorax to the nearest $$\frac{1}{4}$$ inch. How much longer is the walking stick’s abdomen than its thorax? How many times longer is its abdomen than its thorax?

Question 26.
MODELING WITH MATHEMATICS
In 2003, a remote controlled model airplane became the first ever to fly nonstop across the Atlantic Ocean. The map shows the airplane’s position at three different points during its flight. Point A represents Cape Spear. New foundland. point B represents the approximate position after 1 day, and point C represents Mannin Bay’ Ireland. The airplane left from Cape Spear and landed in Mannin Bay.

a. Find the total distance the model airplane flew.

b. The model airplane’s flight lasted nearly 38 hours. Estimate the airplane’s average speed in miles per hour.

Question 27.
USING STRUCTURE
Determine whether the statements are true or False. Explain your reasoning.

a. B is between A and C.
b. C is between B and E.
c. D is between A and H.
d. E is between C and F.

Question 28.
MATHEMATICAL CONNECTIONS
Write an expression for the length of the segment.
a. $$\overline{A C}$$

b. $$\overline{Q R}$$

Question 29.
MATHEMATICAL CONNECTIONS
Point S is between points R and T on $$\overline{R T}$$. Use the information to write an equation in term of x. Then Solve the equation and find RS, ST, and RT.
a. RS = 2x + 10
ST = x – 4
RT = 21

b. RS = 3x – 16
ST = 4x – 8
RT = 60

c. RS = 4x – 9
ST=11
RTx+IO

d. RS = 4x – 9
ST = 19
RT = 8x – 14

Question 30.
THOUGHT PROVOKING
Is it possible to design a table where no two legs have the same length? Assume that the endpoints of the legs must all lie in the same plane. Include a diagram as part o! your answer.

Question 31.
MODELING WITH MATHEMATICS
You have w walk from Room 103 to Room 117.

a. How many feet do you travel from Room 103 to Room 117?
b. You can walk 4.4 feet per second. How many minutes will it take you to get to Room 117?
c. Why might it take you longer than the time in Part (b)?

Question 32.
MAKING AN ARGUMENT
Your friend and your Cousin discuss measuring with a ruler. Your friend says that you must always line up objects at the zero on a ruler. Your cousin says it does not matter. Decide who is correct and explain your reasoning.

Question 33.
REASONING
You travel twin City X to City Y. You know that the round-trip distance is 647 miles. City Z, a city you pass on the way, is 27 miles from City X. Find the distance from City Z to City Y. Justify your answer.

Question 34.
HOW DO YOU SEE IT?
The bar graph shows the win-loss record for a lacrosse team over a period of three years. Explain how you can apply the Ruler Postulate (Post. 1.1) and the Segment Addition Postulate (Post. 1.2) when interpreting a stacked bar graph like the one shown.

Question 35.
ABSTRACT REASONING
The points (a,b) and (c, b) from a segment, and the points (d, e) and (d, f ) from a segment. Create an equation assuming the segments are congruent. Are there any letters not used in the equation? Explain.

Question 36.
MATHEMATICAL CONNECTIONS
In the diagram, $$\overline{A B}$$ ≅ $$\overline{B C}$$, $$\overline{A D}$$ ≅ $$\overline{C D}$$, and AD = 12. Find the lengths of all segments in the diagram. Suppose you choose one of the segments at random. What is the probability that the measure ol the segment is greater than 3? Explain your reasoning.

Question 37.
CRITICAL THINKING
Is it possible to use the Segment Addition Postulate (Post. 1.2) to show FB > CB or that AC > DB? Explain your reasoning.

Maintaining Mathematical Proficiency

Simplify.

Question 38.
$$\frac{-4+6}{2}$$

Question 39.
$$\sqrt{20+5}$$

Question 40.
$$\sqrt{25+9}$$

Question 41.
$$\frac{7+6}{2}$$

Solve the equation.

Question 42.
5x + 7 = 9x – 17

Question 43.
$$\frac{3+y}{2}=6$$ = 6

Question 44.
$$\frac{-5+x}{2}=-9$$ = – 9

Question 45.
– 6x – 13 = – x = 23

### 1.3 Using Midpoint and Distance Formulas

EssentiaI Question
How can you find the midpoint and length of a line segment in a coordinate plane?

Exploration 1

Finding the Midpoint of a Line Segment

Work with a partner.

Use centimeter graph paper.

a.  Graph $$\overline{A B}$$, where the points A and B are as shown.

b. Explain how to bisect $$\overline{A B}$$, that is, to divide AB into two congruent line segments. Then
bisects $$\overline{A B}$$ and use the result to find the midpoint M of $$\overline{A B}$$.

c. What are the coordinates of the midpoint M?

d. Compare the x-coordinates of A, B, and M. Compare the y-coordinates of A, B, and M. How arc the coordinates of the 4 midpoint M related to the coordinates of A and B?

Exploration 2

Finding the Length of a Line Segment work with a partner. Use centimeter graph paper.

b. Use the Pythagorean Theorem to find the length of AB.

c. Use a centimeter ruler to verify the length you found in part (b).
MAKING SENSE OF PROBLEMS
To be proficient in math, you need to check your answers and continually ask yourself, “Does this make sense?’

d. Use the Pythagorean Theorem and point M from Exploration 1 to find the lengths of $$\overline{A M}$$ and $$\overline{M B}$$. What can you conclude?

Question 3.
How can you find the midpoint and length ola line segment in a coordinate plane?

Question 4.
Find the coordinates of the midpoint M and the length of the line segment whose endpoints are given.
a. D(- 10, – 4), E(14, 6)
b. F(- 4, 8), G(9, 0)

### Lesson 1.3 Using Midpoint and Distance Formulas

Monitoring Progress

Identify the segment, bisector of $$\overline{P Q}$$. Then find PQ.

Question 1.

Question 2.

Question 3.
Identify the segment bisector of $$\overline{P Q}$$. Then find MQ.

Question 4.
Identify the segment bisector or $$\overline{R S}$$. Then find RS.

Question 5.
The endpoints of $$\overline{A B}$$ are A (1, 2) and B(7, 8). Find the coordinates of the midpoint M.

Question 6.
The endpoints of $$\overline{C D}$$ are C( – 4, 3) and D(6, 5). Find the coordinates of the midpoint M.

Question 7.
The midpoint of $$\overline{T U}$$ is M(2, 4). One endpoint is T(1, 1). Find the coordinates of endpoint U.

Question 8.
The midpoint of $$\overline{V W}$$ is M (- 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V.

Question 9.
In Example 4, a park is 3 miles east and 4 miles south of your apartment. Find the distance between the park and your school.

### Exercise 1.3 Using Midpoint and Distance Formulas

Question 1.
VOCABULARY
If a point ray, line, line segment or plane intersects a segment at its midpoint, then what does it do to the segment?

Question 2.
COMPLETE THE SENTENCE
To find the length of $$\overline{A B}$$, with endpoints A(- 7, 5) and B(4, – 6). you can use the _________ .

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6. identify the segment bisector of $$\overline{R S}$$. Then find RS.

Question 3.

Question 4.

Question 5.

Question 6.

In Exercises 7 and 8, identify the segment bisector of $$\overline{J K}$$. Then find JM.

Question 7.

Question 8.

In Exercises 9 and 10. identify the segment bisector of $$\overline{X Y}$$. Then find XY.

Question 9.

Question 10.

CONSTRUCTION
In Exercises 11 – 14, copy the segment and construct a segment bisector by paper folding. Then label the midpoint M.

Question 11.

Question 12.

Question 13.

Question 14.

In Exercises 15 – 18, the endpoints of $$\overline{C D}$$ are given. Find the coordinates of the midpoint M.

Question 15.
C (3, – 5) and D (7, 9)

Question 16.
C (4, 7) and D (0, – 3)

Question 17.
C (- 2, 0) and D (4, 9)

Question 18.
C (- 8, – 6) and D (- 4, 10)

In Exercises 19 – 22. the midpoint M and one endpoint of $$\overline{G H}$$ are given. Find the coordinates of the other endpoint.

Question 19.
G (5, – 6) and M (4, 3)

Question 20.
H (- 3, 7) and M (- 2, 5)

Question 21.
H ( – 2, 9) and M(8, 0)

Question 22.
G (- 4, 1) and M (- $$\frac{13}{2}$$, – 6)

In Exercises 23 – 30, find the distance between the two points.

Question 23.
A (13, 2) and B (7, 10)

Question 24.
C (- 6, 5) and D (- 3, 1)

Question 25.
E (3, 7) and F (6, 5)

Question 26.
G (- 5, 4) and H (2, 6)

Question 27.
J (- 8, 0) and K (1, 4)

Question 28.
L (7, – 1) and M (- 2, 4)

Question 29.
R (0, 1) and S (6, 3.5)

Question 30.
T (13, 1.6) and V (5.4, 3.7)

ERROR ANALYSIS
In Exercises 31 and 32, describe and correct the error in finding the distance between A(6, 2) and 8(1, – 4).

Question 31.

Question 32.

COMPARING SEGMENTS
In Exercises 33 and 34, the endpoints of two segments are given. Find each segment length. Tell whether the segments are congruent. If they are not congruent, state which segment length is greater.

Question 33.
$$\overline{A B}$$: A(0, 2), B(- 3, 8) and $$\overline{C D}$$: C(- 2, 2), D(0, – 4)

Question 34.
$$\overline{E F}$$: E(1, 4),F(5, 1) and $$\overline{G H}$$: G(-3, 1), H(1, 6)

Question 35.
WRITING
Your friend is having trouble understanding the Midpoint Formula.
a. Explain how to find the midpoint when given the two endpoints in your own words.
b. Explain how to tind the other endpoint when given one endpoint and the midpoint in your own words.

Question 36.
PROBLEM SOLVING
In baseball, the strike zone is
the region a baseball needs to pass through for the umpire to declare it a strike when the butler does not swing. The top of the strike zone is a horizontal plane passing through the midpoint in the top of the batter’s shoulders and the top of the uniform pants when the player is in a batting stance. Find the height of T. (Note: All heights arc in inches.)

Question 37.
MODELING WITH MATHEMATICS
The figure shows the position of three players during part of a water polo match. Player A throws the ball to Pla’er B. who then throws the ball to Player C.

a. Ho far did Player A throw the hail? Player B?
b. How far would Player A have to throw the ball to throw it directly to Player C?

Question 38.
MODELING WITH MATHEMATICS
Your school is 20 blocks east and 12 blocks south of your house. The mall is 10 blocks north and 7 blocks west of our house. You plan on going to the mall right after school. Find the distance hcLween your school and the mall assuming there is a road directly connecting the school and the mall. One block is 0.1 mile.

Question 39.
PROBLEM SOLVING
A path goes around a triangular park, as shown.

a. Find the distance around the park to the nearest yard.
b. A new path and a bridge are constructed from point Q to the midpoint M of $$\overline{P R}$$. Find QM to the nearest yard.
c. A man jogs from P to Q to M Lo R to Q and back to P at an average speed of 150 yards per minute. About how many minutes does it take? Explain your reasoning.

Question 40.
MAKING AN ARGUMENT
Your friend claims there is an easier way to find the length of a segment than the Distance Formula when the x-coordinates of the endpoints are equal. He claims all you have to do is subtract the y-coordinates. Do you agree with his statement? Explain your reasoning.

Question 41.
MATHEMATICAL CONNECTIONS
Two points are located at (a, c) and (b, c). Find the midpoint and the distance between the two points.

Question 42.
HOW DO YOU SEE IT?
$$\overline{A B}$$ contains midpoint M and points C and D. as shown. Compare the lengths. If you cannot draw a conclusion. write impossible to tell. Explain your reasoning.
a. AM and MB
b. AC and MB
c. MC and MD
d. MB and DB

Question 43.
ABSTRACT REASONING
Use the diagram in Exercise 42. The points on $$\overline{A B}$$ represent locations you pass on your commute to work. You travel from your home at location A to location M before realizing that your left your lunch at home. You could turn around to get your lunch and then continue to work at location B. Or you could go to work and go to location D for lunch today. You want to choose the option that involves the least distance you must travel. which option should you choose? Explain your reasoning.

Question 44.
THOUGHT PROVOKING
Describe three ways to divide a rectangle into two congruent regions. Do the regions have to he triangles? Use a diagram to support your answer.

Question 45.
ANALYZING RELATIONSHIPS
The length of $$\overline{X Y}$$ is 24 centimeters. The midpoint of $$\overline{X Y}$$ is M. and C is on $$\overline{X M}$$ so that XC is $$\frac{2}{3}$$ of XM. Point D is on $$\overline{M Y}$$ so that MD is $$\frac{3}{4}$$ of MY. What is the length of $$\overline{C D}$$?

Maintaining Mathematical Proficiency

Find the perimeter and area of the figure.

Question 46.

Question 47.

Question 48.

Question 49.

Solve the inequality. Graph the solution.

Question 50.
a + 18 < 7

Question 51.
y – 5 ≥ 8

Question 52.
– 3x > 24

Question 53.
$$\frac{z}{4}$$ ≤ 12

### Study Skills: Keeping Your Mind Focused

1.1 – 1.3 What did you learn

Mathematical Practices

Question 1.
Sketch an example of the situation described in Exercise 49 on page 10 in a coordinate plane. Label your figure.

Question 2.
Explain how you arrived at your answer for Exercise 35 on page 18.

Question 3.
What assumptions did you make when solving Exercise 43 0n page 26?

### 1.1 – 1.3 Quiz

Use the diagram

Question 1.
Name four points.

Question 2.
Name three collinear points.

Question 3.
Name two lines.

Question 4.
Name three coplanar points.

Question 5.
Name the plane that is shaded green.

Question 6.
Give two names for the plane that is shaded blue.

Question 7.
Name three line segments.

Question 8.
Name three rays.

Sketch the figure described.

Question 9.
$$\vec{Q}$$R and

Question 10.
plane P intersecting  at Z

Plot the points in a coordinate plane. Then determine whether $$\overline{A B}$$ and $$\overline{C D}$$ are congruent.

Question 11.
A(- 3, 3), B( 1, 3), C(3, 2), D(3, – 2)

Question 12.
A(- 8, 7), B(1, 7), C(- 3, – 6), D(5, – 6)

Find AC. (Section 1.2)

Question 13.

Question 14.

Find the coordinates of the midpoint M and the distance between the two points.

Question 15.
J(4, 3) and K(2, – 3)

Question 16.
L(- 4, 5) and N(5, – 3)

Question 17.
P(- 6, – 1) and Q(1, 2)

Question 18.
Identify the segment bisector of $$\overline{R S}$$. Then find RS.

Question 19.
The midpoint of $$\overline{J K}$$ is M(0, 1). One endpoint is J(- 6, 3). Find the coordinates of endpoint K.

Question 20.
Your mom asks you to run some errands on your way home from school. She wants you to Stop at the post office and the grocery store, which are both on the same straight road between your school and your house. The distance from your school to the post office is 376 yards. the distance from the post office to your house is 929 yards. and the distance from the grocery store to your house is 513 yards.
a. Where should you stop first?

b. What is the distance from the post office to the grocery store?

d. you walk at a speed of 75 yards per minute. How long does it take you to walk straight home from school? Explain your answer.

Question 21.
The figure shows a coordinate plane on a baseball held. The distance from home plate to first base is 90 feet.The pitching mound is the midpoint between home plate and second base. Find the distance from home plate to second base. Find the distance between home plate and the pitching mound. Explain how you hound our answers.

### 1.4 Perimeter and Area in the Coordinate Plane

EssentiaI Question

How can you find the perimeter and area of a polygon in a coordinate plane?

Exploration 1

Finding the Perimeter and Area of a Quadrilateral

Work with a partner

a. On a piece of centimeter graph paper. draw quadrilateral ABCD in a coordinate plane. Label the points A(1, 4), B(- 3, 1) C(0, – 3), and D(4, 0).

b. Find the perimeter of quadrilateral ABCD.

c. Are adjacent sides 01 quadrilateral ABCD perpendicular to each other? How can you tell?

Exploration 2

Finding the Area of a Polygon Work with a partner.

a. Partition quadrilateral ABCD into tour right triangles and one square, as shown. Find the coordinates of the vertices for the five smaller polygons.

b. Find the areas of the five smaller polygons.

c. Is the sum of the areas of the five smaller polygons equal to the area of quadrilateral ABCD? Justify your answer.

Question 3.
How can you find the perimeter and area of a polygon in a coordinate plane?

Question 4.
Repeat Exploration 1 for quadrilateral EFGH. where the coordinates of the vertices are E(- 3, 6), F(- 7, 3), G(- 1, – 5), and H(3, – 2).

### Lesson 1.4 Perimeter and Area in the Coordinate Plane

Monitoring Progress

Classify the polygon the number of sides. Tell whether it is convex or concave.

Question 1.

Question 2.

Find the perimeter of the polygon with the given vertices.

Question 3.
D(- 3, 2), E(4, 2), F(4, – 3)

Question 4.
G(- 3, 2), H(2, 2), J(- 1, – 3)

Question 5.
K( – 1, 1), L(4, 1), M(2, – 2), N(- 3, – 2)

Question 6.
Q(- 4, – 1), R(1, 4), S(4, 1), T(- 1, – 4)

Find the area of the polygon with the given vertices.

Question 7.
G(2, 2), H(3, – 1), J(- 2, – 1)

Question 8.
N(- 1, 1), P(2, 1), Q(2, – 2), R(- 1, – 2)

Question 9.
F(- 2, 3), G(1, 3), H(1, – 1), J(- 2, – 1)

Question 10.
K(- 3, 3), L(3, 3), M(3, – 1), N(- 3, – 1)

Question 11.
You are building a patio in your school’s courtyard. In the diagram at the left, the coordinates represent the four vertices of the patio. Each unit in the coordinate plane represents 1 foot. Find the area of the patio.

### Exercise 1.4 Perimeter and Area in the Coordinate Plane

Question 1.
COMPLETE THE SENTENCE
The perimeter of a square with side length s is P = _________ .

Question 2.
WRITING
What formulas can you use to find the area of a triangle in a coordinate plane?

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6, cIassify the polygon by the number of sides. Tell whether it is convex or concave.

Question 3.

Question 4.

Question 5.

Question 6.

In Exercises 7 – 12. find the perimeter of the polygon with the given vertices.

Question 7.
G(2, 4), H(2, – 3), J(- 2, – 3), K(- 2, 4)

Question 8.
Q(- 3, 2), R(1, 2), s(1, – 2), T(- 3, – 2)

Question 9.
U(- 2, 4), V(3, 4), W(3, – 4)

Question 10.
X(- 1, 3), Y(3, 0), Z(- 1, – 2)

Question 11.

Question 12.

In Exercises 13 – 16. find the area of the polygon with the given vertices.

Question 13.
E(3, 1), F(3, – 2), G(- 2, – 2)

Question 14.
J(- 3, 4), K(4, 4), L(3, – 3)

Question 15.
W(0, 0), X(0, 3), Y(- 3, 3), Z(- 3, 0)

Question 16.
N(- 2, 1), P(3, 1), Q(3, – 1), R(- 2, 1)

In Exercises 17 – 24, use the diagram.

Question 17.
Find the perimeter of △CDE.

Question 18.
Find the perimeter of rectangle BCEF.

Question 19.
Find the perimeter of △ABF.

Question 20.
Find the perimeter of quadrilateral ABCD.

Question 21.
Find the area of △CDE.

Question 22.
Find the area of rectangle BCEF

Question 23.
Find the area of △ABF

Question 24.
Find the area of quadrilateral ABCD.

ERROR ANALYSIS
In Exercises 25 and 26, describe and correct the error in finding the perimeter or area of the polygon.

Question 25.

Question 26.

In Exercises 27 and 28, use the diagram.

Question 27.
Determine which point is the remaining vertex of a triangle with an area of 4 square units.
A. R(2, 0)
B. S(- 2, – 1)
C. T(- 1, 0)
D. U(2, – 2)

Question 28.
Determine which points are the remaining vertices of a rectangle with a perimeter of 14 units.
A. A(2, – 2) and B(2, – 1)
B. C(- 2, – 2) and D(- 2, 2)
C. E(- 2, – 2) and F(2, – 2)
D. G(2, 0) and H(2, 0)

Question 29.
USING STRUCTURE
Use the diagram.

a. Find the areas of square EFGH and square EJKL. What happens to the area when the perimeter of square EFGH is doubled?
b. Is this true for every square? Explain.

Question 30.
MODELING WITH MATHEMATICS
You are growing zucchini plants in your garden. In the figure. the entire garden is rectangle QRST. Each unit in the coordinate plane represents 1 foot.

a. Find the area of the garden.
b. Zucchini plants require 9 square feet around each plant. How many zucchini plants can you plant?c. You decide to use square TUVWto grow lettuce. you can plant lour heads of lettuce per square loot. How man of each vegetable can you plant? Explain.

Question 31.
MODELING WITH MATHEMATICS
You are going for a hike in the woods. You hike to a water fall that is 4 miles east of where you left your car. You then hike to a lookout point that is 2 miles north of your car. From the lookout point. you return to our ear.

a. Map out your route in a coordinate plane with our car at the origin. Let each unit in the coordinate plane represent 1 mile. Assume you travel along straight paths.
b. How far do you travel during the entire hike?
c. When you leave the waterfall, you decide to hike to an old wishing well before going to the lookout point. The wishing well is 3 miles north and 2 miles west of the lookout point. How far do you travel during the entire hike?

Question 32.
HOW DO YOU SEE IT?
Without performing any calculations, determine whether the triangle or the rectangle has a greater area. Which one has a greater perimeter? Explain your reasoning.

Question 33.
MATHEMATICAL CONNECTIONS
The lines y1 = 2x – 6, y2 = – 3x + 4, and y3 = – $$\frac{1}{2}$$ + 4 are the sides of a right triangle.
a. Use slopes to determine which sides are perpendicular.
b. Find the vertices of the triangle.
c. Find the perimeter and area of the triangle.

Question 34.
THOUGHT PROVOKING
Your bedroom has an area of 350 square feet. You are remodeling to include an attached bathroom that has an area of 150 square feet. Draw a diagram of the remodeled bedroom and bathroom in a coordinate plane.

Question 35.
PROBLEM SOLVING
Use the diagram

a. Find the perimeter and area of the square.
b. Connect the midpoints of the sides of the given square to make a quadrilateral. Is this quadrilateral a square? Explain our reasoning.
c. Find the perimeter and area of the quadrilateral you made in part (b). Compare this area to the area you found in part (a).

Question 36.
MAKING AN ARGUMENT
Your friend claims that a rectangle with the same perimeter as △QRS will have the same area as the triangle. Is your friend correct? Explain your reasoning.

Question 37.
REASONING
Triangle ABC has a perimeter of 12 units. The vertices of the triangle are A(x, 2), B(2, – 2), and C(- 1, 2). Find the value of x.

Maintaining Mathematical Proficiency

Solve the equation.

Question 38.
3x – 7 = 2

Question 39.
5x + 9 = 4

Question 40.
x + 4 = x – 12

Question 41.
4x- 9 = 3x + 5

Question 42.
11 – 2x = 5x – 3

Question 43.
$$\frac{x+1}{2}$$ = 4x – 3

Question 44.
Use a compass and straightedge to construct a copy of the line segment.

### 1.5 Measuring and Constructing Angles

Essential Question

How can you measure and classify an angle?

Exploration 1

Measuring and Classifying Angles

Work with a partner: Find the degree measure of each of the following angles. Classify each angle as acute, right, or obtuse.

a. ∠AOB
b. ∠AOC
c. ∠BOC
d. ∠BOE
e. ∠COE
f. ∠COD
g. ∠BOD
h. ∠AOE

Exploration 2

Drawing a Regular Polygon

Work with a partner.

a. Use a ruler and protractor to draw the triangular pattern shown at the right.

b. Cut out the pattern and use it to draw three regular hexagons. as shown below.

C. The sum of the angle measures of a polygon with n sides is equal to 180(n – 2)°. Do the angle measures et your hexagons agree with this rule? Explain.
ATTENDING TO PRECISION
To be proficient in math, you need to calculate and measure accurately and efficiently.

d. Partition your hexagons into smaller polygons. as shown below. For each hexagon. find the sum et the angle measures of the smaller polygons Does each sum equal the sum of the angle measures o1 a hexagon? Explain.

Question 3.
How can you measure and classify an angle?

### Lesson 1.5 Measuring and Constructing Angles

Monitoring Progress

Write three names for the angle.

Question 1.

Question 2.

Question 3.

Use the diagram in Example 2 to find the angle measure. Then classify the angle.

Question 4.
∠JHM

Question 5.
∠MHK

Question 6.
∠MHL

Question 7.
Without measuring, is ∠DAB ≅ ∠FEH in Example 3? Explain your reasoning. Use a protractor to verify your answer.

Find the indicated angle measures.

Question 8.
Given that ∠KLM is a straight angle. find in ∠KLN and, m∠NLM.

Question 9.
Given that ∠EFG is a right angle. find, m∠EFH and m∠HFG.

Question 10.
Angle MNP is a straight angle, and $$\vec{N}$$Q bisects ∠MNP. Draw ∠MNP and $$\vec{N}$$Q. Use arcs to mark the congruent angles in your diagram. Find the angle measures of these congruent angles.

### Exercise 1.5 Measuring and Constructing Angles

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
Two angles are __________ angles when they have the same measure.

Question 2.
WHICH ONE DOESN’T BELONG?
Which angle name does not belong with the other three? Explain your reasoning.

∠BCA
∠BAC
∠1
∠CAB

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6. write three names for the angle.

Question 3.

Question 4.

Question 5.

Question 6.

In Exercises 7 and 8, name three different angles in the diagram.

Question 7.

Question 8.

In Exercises 9 – 12 find the angle measure. Then classify the angle.

Question 9.
m∠AOC

Question 10.
m∠BOD

Question 11.
m∠COD

Question 12.
m∠EOD

ERROR ANALYSIS
In Exercises 13 and 14. describe and correct the error in finding the angle measure. Use the diagram from Exercises 9 – 12.

Question 13.

Question 14.

CONSTRUCTION
In Exercises 15 and 16. use a compass and straightedge to copy the angle.

Question 15.

Question 16.

In Exercises 17 – 20, in m∠AED = 34° and m∠EAD = 112°.

Question 17.
Identify the angles congruent to ∠IED.

Question 18.
Identify the angles congruent to ∠EAD.

Question 19.
Find m∠BDC.

Question 20.

In Exercises 21 – 24, find the indicated angle measure.

Question 21.
Find m∠ABC.

Question 22.
Find m∠LMN.

Question 23.
m∠RST = 114°. Find m∠RSV.

Question 24.
∠GHK is a straight angle. Find m∠LHK.

In Exercises 25 – 30. find the indicated angle measures.

Question 25.
m∠ABC = 95°. Find m∠ABD and m∠DBC.

Question 26.
m∠XYZ = 117°. Find m∠XYW and m∠WYZ

Question 27.
∠LMN is a straight angle. Find m∠LMP and m∠NMP

Question 28.
∠ABC is a straight angle. Find m∠ABX and m∠CBX.

Question 29.
Find m∠RSQ and M∠TSQ.

Question 30.
Find m∠DEH and m∠FEH.

CONSTRUCTION
In Exercises 31 and 32. copy the angle. Then construct the angle bisector with a compass and straightedge.
Question 31.

Question 32.

In Exercises 33 – 36, $$\vec{Q}$$S bisects ∠PQR. Use the diagram and the given angle measure to find the indicated angle measures.

Question 33.
m∠PQS = 63°. Find m∠RQS and m∠PQR.

Question 34.
m∠RQS = 71°. Find m∠PQS and m∠PQR.

Question 35.
m∠PQR = 124°. Find m∠PQS and m∠RQS.

Question 36.
m∠PQR = 119°. Find m∠PQS and m∠RQS.

In Exercises 37 – 40, $$\overrightarrow{B D}$$ bisects ∠ABC. Find m∠ABD, m∠CBD, and m∠ABC.

Question 37.

Question 38.

Question 39.

Question 4o.

Question 41.
WRITING
Explain how to find m∠ABD when you are given m∠ABC and m∠CBD.

Question 42.
ANALYZING RELATIONSHIPS
The map shows the intersections of three roads. Malcom Way intersects Sydney Street at an angle of 162°. Park Road intersects Sydney Street at an anÌe of 87°. Find the angle at which Malcom Way intersects Park Road.

Question 43.
ANALYZING RELATIONSHIPS
In the sculpture shown in the photograph. the measure of ∠LMN is 76° and the measure of ∠PMN is 36°. What is the measure of ∠LMP?

USING STRUCTURE
In Exercise 44 – 46. use the diagram of the roof truss.

Question 44.
In the roof truss, $$\vec{B}$$G bisects ∠ABC and ∠DEF. m∠ABC = 112°. and ∠ABC ≅ ∠DEF Find the measure of each angle.
a. m∠DEF
b. m∠ABG
c. m∠CBG
d. m∠DEG

Question 45.
In the roof truss, ∠DGF is a straight angle and $$\vec{G}$$B bisects ∠DGF Find m∠DGE and m∠FGE.

Question 47.
Name an example of each of the four types of angles according to their measures in the diagram.

MATHEMATICAL CONNECTIONS
In ∠ABC. $$\vec{B}$$X is in the interior 0f the angle. m∠ABX is 12 more than 4 times m∠CBX. and in m∠ABC = 92°.
a. Draw a diagram to represent the situation.
b. Write and solve an equation to find m∠ABX and, m∠CBX.

Question 48.
THOUGHT PROVOKING
The angle between the minute hand and the hour hand of a clock is 90° What time is it? Justify your answer.

Question 49.
ABSTRACT REASONING
Classify the angles that result from bisecting each type of angle.
a. acute angle
b. right angle
c. obtuse angle
d. straight angle

Question 50.
ABSTRACT REASONING
Classify the angles that result from drawing a ray in the interior of each type of angle. Include all possibilities and explain your reasoning.
a. acute angle
b. right angle
c. obtuse angle
d. straight angle

Question 51.
CRITICAL THINKING
The ray from the origin through (4, 0) form one side of an angle. Use the numbers below as x- and y-coordinates to create each type of angle in a coordinate plane.
– 2 – 1 0 1 2
a. acute angle
b. right angle
c. obtuse angle
d. straight angle

Question 52.
MAKING AN ARGUMENT
Your friend claims it is possible for a straight angle to Consist of two obtuse angles. Is your friend correct? Explain your reasoning.

Question 53.
CRITICAL THINKING
Two acute angles are added together. What type(s) of angle(s) do they form? Explain your reasoning.

Question 54.
HOW DO YOU SEE IT?
Use the diagram

a. Is it possible for ∠XYZ to be a straight angle? Explain your reasoning.

b. What can you change in the diagram so that ∠XYZ is a straight angle?

Question 55.
WRITING
Explain the process of bisecting an angle in your own words. Compare it to bisecting a segment.

Question 56.
ANALYZING RELATIONSHIPS
$$\vec{S}$$Q bisects ∠RST, $$\vec{S}$$P bisects ∠RSQ. and $$\vec{S}$$V bisects ∠RSP. The measure of ∠VSP is 17°. Find in m∠TSQ. Explain.

Question 57.
ABSTRACT REASONING
A bubble level is a tool used
to determine whether a surface is horizontal. like the top of a picture frame. If the bubble is not exactly in the middle when the level is placed on the surface. then the surface is not horizontal. What is the most realistic type of angle Formed by the level and a horizontal line when the bubble is not in the middle? Explain your reasoning.

Maintaining Mathematical Proficiency

Solve the equation.

Question 58.
x + 67 = 180

Question 59.
x + 58 = 90

Question 60.
16 + x = 90

Question 61.
109 + x = 180

Question 62.
(6x + 7) + (13x + 21) = 180

Question 63.
(3x + 15) + (4x – 9) = 90

Question 64.
(11x – 25) + (24x + 10) = 90

Question 65.
(14x – 18) + (5x + 8) = 180

### 1.6 Describing Pairs of Angles

Essential Question

How can you describe angle pair relationships and use these descriptions to find angle measures?

Exploration 1

Work with a partner: The five-pointed star has a regular pentagon at its center.

a. What do you notice about the following angle pairs?
x° and y°
y° and z°
x° and z°

b. Find the values of the indicated variables. Do not use a protractor to measure the angles.

Explain how you obtained each answer.

Exploration 2

Finding Angle Measures

Work with a partner: A square is divided by its diagonals into four triangles.

a. What do you notice about the following angle pairs?
a° and b°
c° and d°
c° and e°

b. Find the values of the indicated variables. Do not use a protractor to measure the angles.

Explain how you obtained each answer.

Question 3.
How can you describe angle pair relationships and use these descriptions to find angle measures?

Question 4.
What do you notice about the angle measures of complementary angles. supplementary angles, and vertical angles?
ATTENDING TO PRECISION
To be proficient in math, you need to communicate precisely with others.

### Lesson 1.6 Describing Pairs of Angles

Monitoring Progress

In Exercises 1 and 2. use the figure.

Question 1.
Name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

Question 2.
Are ∠KGH and ∠LKG adjacent angles? Are ∠FGK and ∠FGH adjacent angles? Explain.

Question 3.
∠1 is a complement of ∠2. arid m∠2 = 5°. Find m∠1.

Question 4.
∠3 is a supplement of ∠4. and m∠3 = 148°. Find m∠4.

Question 5.
∠LMN and ∠PQR are complementary angles. Find the measures of the angles when in m∠LMN = (4x – 2)° and m∠PQR = (9x + 1)°.

Question 6.
Do any of the numbered angles in the figure form a linear pair? Which angles are vertical angles? Explain your reasoning.

Question 7.
The measure of an angle is twice the measure of its complement. Find the measure of each angle.

Question 8.
Two angles form a linear pair. The measure of one angle is 1$$\frac{1}{2}$$ times the measure of the other angle. Find the measure of each angle.

### Exercise 1.6 Describing Pairs of Angles

Vocabulary and Core Concept Check

Question 1.
WRITING
Explain what is different between adjacent angles and vertical angles.

Question 2.
WHICH ONE DOESN’T BELONG?
Which one does hot belong with the other three? Explain your reasoning.

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, use the figure.

Question 3.
Name a pair of adjacent complementary angles.

Question 4.
Name a pair of adjacent supplementary angles.

Question 5.
Name a pair of nonadjacent complementary angles.

Question 6.
Name a pair of nonadjacent supplementary angles.

In Exercises 7 – 10. find the angle measure.

Question 7.
∠1 is a complement of ∠2, and m∠1 = 23°. Find, m∠2.

Question 8.
∠3 is a complement of ∠4. and m∠3 = 46°. Find, m∠4.

Question 9.
∠5 is a supplement of ∠6. and m∠5 = 78°. Find m∠6.

Question 10.
∠7 is a supplement of ∠8. and m∠7 = 109°. Find m∠8.

In Exercises 11 – 14. find the measure of each angle.

Question 11.

Question 12.

Question 13.
∠UVW and ∠XYZ arc complementary angles, m∠UVW = (x – 10)°. and m∠XYZ = (4x – 10)°.

Question 14.
∠EFG and ∠LMN are supplementary angles. m∠EFG = (3x + 17)°, and m∠LMN = ($$\frac{1}{2} x$$ – 5)°

In Exercises 15 – 18. use the figure.

Question 15.
Identify the linear pair(s) that include ∠1.

Question 16.
Identify the linear pair(s) that include ∠7.

Question 17.
Are ∠6 and ∠8 vertical angles? Explain your reasoning.

Question 18.
Are ∠2 and ∠5 vertical angles? Explain your reasoning.

In Exercises 19 – 22, find the measure of each angle.

Question 19.
Two angles from a linear pair. The measure of one angle is twice the measure of the other angle.

Question 20.
Two angles from a linear pair. The measure of one angle is $$\frac{1}{3}$$ the measure of the other angle.

Question 21.
The measure of an angle is nine times the measure of its complement.

Question 22.
The measure of an angle is $$\frac{1}{4}$$ the measure of its complement.

ERROR ANALYSIS
In Exercises 23 and 24, describe and correct the error in identifying pairs of angles in the figure.

Question 23.

Question 24.

In Exercises 25 and 26. the picture shows the Alamillo Bridge in Seville, Spain. In the picture, m∠1 = 58° and m∠2 = 24°.

Question 25.
Find the measure of the supplement of ∠1

Question 26.
Find the measure of the supplement of ∠26.

Question 27.
PROBLEM SOLVING
The arm of a crossing gate moves 42° from a vertical position. How many more degrees does the arm have to move so that it is horizontal?

A. 42°
B. 138°
C. 48°
D. 90°

Question 28.
REASONING
The foul lines of a baseball field intersect at home plate to form a right angle. A batter hits a fair ball such that the path of the baseball forms an angle of 27° with the third base foul line. What is the measure of the angle between the first base foul line and the path of the baseball?

Question 29.
CONSTRUCTION
Construct a linear pair where one angle measure is 115°.

Question 30.
CONSTRUCTION
Construct a pair of adjacent angles that have angle measures of 45° and 97°.

Question 31.
PROBLEM SOLVING
m∠U = 2x°, and m∠V = 4m∠U. Which value of x makes ∠U and ∠V complements of each other?
A. 25
B. 9
C. 36
D. 18

MATHEMATICAL CONNECTIONS
In Exercises 32 – 35, write and solve an algebraic equation to find the measure of each angle based on the given description.

Question 32.
The measure of an angle is 6° less than the measure of its complement.

Question 33.
The measure of an angle is 12° more than twice the measure of its complement.

Question 34.
The measure of one angle is 3° more than $$\frac{2}{3}$$ the measure of its supplement.

Question 35.
Two angles form a linear pair. The measure of one angle is 15° less than $$\frac{2}{3}$$ the measure of the other angle.

CRITICAL THINKING
In Exercises 36 – 41. tell whether the statement is always, sometimes, or never true. Explain your reasoning.

Question 36.

Question 37.
Angles in a linear pair are supplements of each other.

Question 38.

Question 39.
Vertical angles are supplements of each other.

Question 40.
If an angle is acute. then its complement is greater than its supplement.

Question 41.
If two complementary angles are congruent, then the measure of each angle is 45°,

Question 42.
WRITING
Explain why the supplement of an acute angle must be obtuse.

Question 43.
WRITING
Explain why an obtuse angle does not have a complement.

Question 44.
THOUGHT PROVOKING
Sketch an intersection of roads. Identify any supplementary, complementary, or vertical angles.

Question 45.
ATTENDING TO PRECISION
Use the figure.

a. Find m∠UWV, m∠TWU, and m∠TWX.
b. YOU write the measures of ∠TWU, ∠TWX, ∠UWV and ∠VWX on separate pieces of paper and place the pieces of paper in a box. Then you pick two pieces of paper out of the box at random. What is the probability that the angle measures you choose are supplementary? Explain your reasoning.

Question 46.
HOW DO YOU SEE IT?
Tell whether you can conclude that each statement is true based on the figure. Explain your reasoning.

a. $$\overline{C A}$$ ≅ $$\overline{A F}$$
b. Points C, A, and F are collinear
d. $$\overline{B A}$$ ≅ $$\overline{A E}$$
e.
f. ∠BAC and ∠CAD are complementary angles.
g. ∠DAE is a right angle.

Question 47.
REASONING
∠KJL and ∠LJM arc complements, and ∠MJN and ∠LJM are complements. Can you show that ∠KJL ≅ ∠MJN? Explain your reasoning.

Question 48.
MAKING AN ARGUMENT
Light from a flashlight strikes a mirror and is reflected so that the angle of reflection is congruent to the angle of incidence. Your classmate claims that ∠QPR is congruent to ∠TPU regardless of the measure of ∠RPS. Is your classmate correct? Explain your reasoning.

Question 49.
DRAWING CONCLUSIONS
Use the figure.

a. Write expressions tor the measures of ∠BAE, ∠DAE, and ∠CAB.
b. What do you notice about the measures of vertical angles? Explain your reasoning.

Question 50.
MATHEMATICAL CONNECTIONS
Let m∠1 = x°, m∠2 = y1°, and m∠3 = y2° ∠2 is the complement of ∠1, and ∠3 is the supplement of ∠1.

a. Write equations for y1 as a function of x and for y2 as a function of x. What is the domain of each function? Explain.
b. Graph each function and describe its range.

Question 51.
MATHEMATICAL CONNECTIONS
The sum of the measures of two complementary angles is 74° greater than the difference of their measures. Find the measure of each angle. Explain how you found the angle measures.

Maintaining MathematicaI Proficiency

Determine whether the statement is always, sometimes, or never true. Explain your reasoning.

Question 52.
An integer is a whole number.

Question 53.
An integer is an irrational number.

Question 54.
An irrational number is a real number.

Question 55.
A whole number is negative.

Question 56.
A rational number is an integer.

Question 57.
A natural number is an integer.

Question 58.
A whole number is a rational number.

Question 59.
An irrational number is negative.

### 1.4 – 1.6 Performance Task: Comfortable Horse Stalls

Mathematical Practices

Question 1.
How could you explain your answers to Exercise 33 on page 36 to a friend who is unable to hear’?

Question 2.
What tool(s) could you use to verify your answers to Exercises 25 – 30 on page 44?

Question 3.
Your friend says that the angles in Exercise 28 on page 53 are supplementary angles. Explain why you agree or disagree.

### Basics of Geometry Chapter Review

#### 1.1 Points, Lines, and Planes

Use the diagram.

Question 1.
Give another name for plane M.

Question 2.
Name a line in the plane.

Question 3.
Name a line intersecting the plane.

Question 4.
Name two rays.

Question 5.
Name a pair of opposite rays.

Question 6.
Name a point not in plane M.

#### 1.2 Measuring and Constructing Segments

Find XZ.

Question 7.

Question 8.

Question 9.
Plot A(8, – 4), B(3, – 4), C(7, 1), and D(7, – 3) in a coordinate plane.
Then determine whether $$\overline{A B}$$ and $$\overline{C D}$$ are congruent.

#### 1.3 Using Midpoint and Distance Formulas

Find the coordinates of the midpoint M. Then find the distance between points S and T.

Question 10.
S(- 2, 4) and T(3, 9)

Question 11.
S(6, – 3) and T(7, – 2)

Question 12.
The midpoint of $$\overline{J K}$$ is M(6, 3). One endpoint is J(14, 9). Find the coordinates of endpoint K.

Question 13.
Point M is the midpoint of $$\overline{A B}$$ here AM = 3x + 8 and MB = 6x – 4. Find AB.

#### 1.4 Perimeter and Area in the Coordinate Plane

Find the perimeter and area of the polygon with the given vertices.

Question 14.
W(5, – 1), X(5, 6), Y(2, 1) Z(2, 6)

Question 15.
E(6, – 2) , F(6, 5), G(- 1, 5)

#### 1.5 Measuring and Constructing Angles

Find m∠ABD and m∠CBD.

Question 16.
m∠ABC = 77°

Question 17.
m∠ABC = 111°

Question 18.
Find the measure of the angle using a protractor.

#### 1.6 Describing Pairs of Angles

∠1 and ∠2 are complementary angles. Given m∠1, find m∠2.

Question 19.
m∠1 = 12°

Question 20.
m∠1 = 83°

∠3 and ∠4 are supplementary angles. Given m∠3, find m∠4.

Question 21.
m∠3 = 116°

Question 22.
m∠3 = 56°

### Basics of Geometry Chapter Test

Find the length of $$\overline{Q S}$$. Explain how you found your answer.

Question 1.

Question 2.

Find the coordinates of tile midpoint M. Then find the distance between the two points.

Question 3.
A(- 4, – 8) and B(- 1, 4)

Question 4.
C(- 1, 7) and D(- 8, – 3)

Question 5.
The midpoint of $$\overline{E F}$$ is M(1, – 1). One endpoint is E(- 3, 2). Find the coordinates of
endpoint F.

Use the diagram to decide whether the statement is true or false.

Question 6.
Points A, R, and B are collinear.

Question 7.

Question 8.

Question 9.
Plane D could also be named plane ART.

Find the perimeter and area of the polygon with the gien ertices. Explain how you found your answer.

Question 10.
P(- 3, 4), Q(1, 4), R(- 3, – 2), S(3, – 2)

Question 11.
J(- 1, 3), K(5, 3), L(2, – 2)

Question 12.
In the diagram. ∠AFE is a straight angle and ∠CFE is a right angle. Identify all supplementary and complementary angles. Explain. Then find, m∠DFE, m∠BFC, and m∠BFE.

Question 13.
Use the clock at the left.

a. What is the measure of the acute angle created when the clock is at 10:00?

b. What is the measure of the obtuse angle created when the clock is at 5:00?

c. Find a time where the hour and minute hands create a straight angle.

Question 14.
Sketch a figure that contains a plane and two lines that intersect the plane at one point.

Question 15.
Your parents decide they would like to install a rectangular swimming pool in the backyard. There is a 15-foot by 20-foot rectangular area available. Your parents request a 3-foot edge around each side of the pool. Draw a diagram of this situation in a coordinate plane. What is the perimeter and area of the largest swimming pool that will fit?

Question 16.
The picture shows the arrangement of b0alls in a game of boccie. The object of the game is to throw your ball closest to the small. while ball, which is called the pallino. The green ba1l is the midpoint between the red ball and the pallino. The distance between the green hail and the red ball is 10 inches. The distance between the yellow ball and the pallino is 8 inches. Which bail is closer to the pallino. the green ball or the yellow ball? Explain.

### Basics of Geometry Cumulative Assessment

Question 1.
Use the diagram to determine which segments, if any, are congruent. List all congruent segments.

Question 2.
Order the terms so that each consecutive term builds oft the previous term.
plane           segment      line     point    ray

Question 3.
The endpoints of a line segment are (- 6, 13) and (11, 5). Which choice Shows the correct midpoint and distance between these two points?
(A) $$\left(\frac{5}{2}, 4\right)$$; 18.8 units
(B) $$\left(\frac{5}{2}, 9\right)$$; 18.8 units
(C) $$\left(\frac{5}{2}, 4\right)$$; 9.4 units
(D) $$\left(\frac{5}{2}, 9\right)$$; 9.4 units

Question 4.
Find the perimeter and area of the figure shown

Question 5.
Plot the points W(- 1, 1), X(5, 1), Y(5, – 2), and Z(- 1, – 2) in a coordinate plane. What type of polygon do the points form? Your friend claims that you could use this figure to represent a basketball court with an area of 4050 square feet and a perimeter of 270 feet. Do you support your friend’s claim? Explain.

Question 6.
Use the steps in the construction to explain how you know that $$\vec{A}$$G is the angle bisector
of ∠CAB.

Question 7.
The picture shows an aerial view of a city. Use the streets highlighted in red to identify all congruent angles. Assume all streets are straight angles.