# Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles

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## Big Ideas Math Book Geometry Answer Key Chapter 6 Relationships Within Triangles

### Relationships Within Triangles Maintaining Mathematical Proficiency

Write an equation of the line passing through point P that is perpendicular to the given line.

Question 1.
P(3, 1), y = $$\frac{1}{3}$$x – 5

Question 2.
P(4, – 3), y = – x – 5

Question 3.
P(- 1, – 2), y = – 4x + 13

Write the sentence as an inequality.

Question 4.
A number w is at least – 3 and no more than 8.

Question 5.
A number m is more than 0 and less than 11.

Question 6.
A number s is less than or equal to 5 or greater than 2.

Question 7.
A number d is fewer than 12 or no less than – 7.

Question 8.
ABSTRACT REASONING
Is it possible for the solution of a compound inequality to be all real numbers? Explain your reasoning.

### Relationships Within Triangles Mathematical Practices

Monitoring Progress

Refer to the figures at the top of the page to describe each type of line, ray, or segment in a triangle.

Question 1.
perpendicular bisector

Question 2.
angle bisector

Question 3.
median

Question 4.
altitude

Question 5.
midsegment

### 6.1 Perpendicular and Angle Bisectors

Exploration 1

Points on a Perpendicular Bisector

Work with a partner. Use dynamic geometry software.

a. Draw any segment and label it $$\overline{A B}$$. ConStruct the perpendicular bisector of $$\overline{A B}$$.

b. Label a point C that is on the perpendicular bisector of $$\overline{A B}$$ but is not on $$\overline{A B}$$.

c. Draw $$\overline{C A}$$ and $$\overline{C B}$$ and find their lengths. Then move point C to other locations on the perpendicular bisector and note the lengths of $$\overline{C A}$$ and $$\overline{C B}$$.

d. Repeat parts (a) – (c) with other segments. Describe any relationships(s) you notice.

Exploration 2

Points on an Angle Bisector

Work with a partner. Use dynamic geometry software.

a. Draw two rays $$\vec{A}$$B and $$\vec{A}$$C to form ∠BAC. Construct the bisector of ∠BAC.

b. Label a point D on the bisector of ∠BAC.

c. Construct and find the lengths of the perpendicular segments from D to the sides of ∠BAC. Move point D along the angle bisector and note how the lengths change.

d. Repeat parts (a)-(c) with other angles. Describe an relationship(s) you notice.
USING TOOLS STRATEGICALLY
To be proficient in math, you need to visualize the results of varying assumptions, explore consequences, and compare predictions with data.

Question 3.
What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle?

Question 4.
In Exploration 2. what is the distance from point D to $$\vec{A}$$B when the distance from D to $$\vec{A}$$C is 5 units? Justify your answer.

### Lesson 6.1 Perpendicular and Angle Bisectors

Use the diagram and tile given information to find the indicated measure.

Question 1.
is the perpendicular bisector of $$\overline{W Y}$$, and $$\overline{y Z}$$ = 13.75. Find WZ.

Question 2.
is the perpendicular bisector of $$\overline{W Y}$$, WZ = 4n – 13, and YZ = n + 17. Find YZ.

Question 3.
Find WX when WZ = 20.5. WY = 14.8. and YZ = 20.5.

Use the diagram and the given information to find the indicated measure.

Question 4.
$$\vec{B}$$D bisects ∠ABC, and DC = 6.9, Find DA.

Question 5.
$$\vec{B}$$D bisects ∠ABC, AD = 3z + 7, and CD = 2z + 11. Find CD.

Question 6.
Find m∠ABC when AD = 3.2, CD = 3.2, and m∠DBC = 39°.

Question 7.
Do you have enough information to conclude that $$\vec{Q}$$S bisects ∠PQR? Explain.

Question 8.
Write an equation of the perpendicular bisector of the segment with endpoints (- 1, – 5) and (3, – 1).

### Exercise 6.1 Perpendicular and Angle Bisectors

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
Point C is in the interior of ∠DEF. If ∠DEC and ∠CEF are congruent, then $$\vec{E}$$C is the ________ of ∠DEF.

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

Is point B the same distance from both X and Z?

Is point B equidistant from X and Z?

Is point B collinear with X and Z?

Is point B on the perpendicular bisector of $$\overline{X Z}$$?

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, find the indicated measure. Explain your reasoning.

Question 3.
GH

Question 4.
QR

Question 5.
AB

Question 6.
UW

In Exercises 7-10. tell whether the information in the diagram allows you to conclude that point P lies on the perpendicular bisector of $$\overline{L M}$$. Explain your reasoning.

Question 7.

Question 8.

Question 9.

Question 10.

In Exercises 11-14. find the indicated measure. Explain your reasoning.

Question 11.
m∠ABD

Question 12.

Question 13.
m∠KJL

Question 14.
FG

In Exercises 15 and 16, tell whether the information in the diagram allows you to conclude that $$\vec{E}$$H bisects ∠FEG. Explain your reasoning.

Question 15.

Question 16.

In Exercises 17 and 18, tell whether the information in the diagram allows you to conclude that DB = DC. Explain your reasoning.

Question 17.

Question 18.

In Exercises 19-22, write an equation of the perpendicular bisector of the segment with the given endpoints.

Question 19.
M(1, 5), N(7, – 1)

Question 20.
Q(- 2, 0), R(6, 12)

Question 21.
U(- 3, 4), V(9, 8)

Question 22.
Y( 10, – 7), Z(- 4, 1)

ERROR ANALYSIS
In Exercises 23 and 24, describe and correct the error in the student’s reasoning.

Question 23.

Question 24.

Question 25.
MODELING MATHEMATICS
In the photo, the road is perpendicular to the support beam and $$\overline{A B} \cong \overline{C B}$$. Which theorem allows you to conclude that $$\overline{A D} \cong \overline{C D}$$?

Question 26.
MODELING WITH MATHEMATICS
The diagram shows the position of the goalie and the puck during a hockey game. The goalie is at point G. and the puck is at point P.

a. What should be the relationship between $$\vec{P}$$G and ∠APB to give the goalie equal distances to travel on each side of $$\vec{P}$$G?

b. How does m∠APB change as the puck gets closer to the goal? Does this change make it easier or more difficult for the goalie to defend the goal? Explain your reasoning.

Question 27.
CONSTRUCTION
Use a compass and straightedge to construct a copy of $$\overline{X Y}$$. Construct a perpendicular bisector and plot a point Z on the bisector so that the distance between point Z and $$\overline{X Y}$$ is 3 centimeters. Measure $$\overline{X Z}$$ and $$\overline{Y Z}$$. Which theorem does this construction demonstrate?

Question 28.
WRITING
Explain how the Converse of the Perpendicular Bisector Theorem (Theorem 6.2) is related to the construction of a perpendicular bisector.

Question 29.
REASONING
What is the value of x in the diagram?

(A) 13
(B) 18
(C) 33
(D) not enough information

Question 30.
REASONING
Which point lies on the perpendicular bisector of the segment with endpoints M(7, 5) and m(- 1, 5)?
(A) (2, 0)
(B) (3, 9)
(C) (4, 1)
(D) (1, 3)

Question 31.
MAKING AN ARGUMENT
Your friend says it is impossible for an angle bisector of a triangle to be the same line as the perpendicular bisector of the opposite side. Is your friend correct? Explain your reasoning.

Question 32.
PROVING A THEOREM
Prove the Converse of the Perpendicular Bisector Theorem (Thm. 6.2). (Hint: Construct a line through point C perpendicular to $$\overline{A B}$$ at point P.)

Given CA = CB
Prove Point C lies on the perpendicular bisector of $$\overline{A B}$$.

Question 33.
PROVING A THEOREM
Use a congruence theorem to prove each theorem.
a. Angle Bisector Theorem (Thin. 6.3)
b. Converse of the Angle Bisector Theorem (Thm. 6.4)

Question 34.
HOW DO YOU SEE IT?
The figure shows a map of a city. The city is arranged so each block north to south is the same length and each block east to west is the same length.

a. Which school is approximately equidistant from both hospitals? Explain your reasoning.

b. Is the museum approximately equidistant from Wilson School and Roosevelt School? Explain your reasoning.

Question 35.
MATHEMATICAL CONNECTIONS
Write an equation whose graph consists of all the points in the given quadrants that are equidistant from the x- and y-axes.
a. I and III
b. II and IV
c. I and II

Question 36.
THOUGHT PROVOKING
The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, is it possible for two lines to be perpendicular but not bisect each other? Explain your reasoning.

Question 37.
PROOF
Use the information in the diagram to prove that $$\overline{A B} \cong \overline{C B}$$ if and onI if points D, E, and B are collinear.

Question 38.
PROOF
prove the statement in parts (a) – (c)

Given Plane P is a perpendicular bisector of $$\overline{X Z}$$ at point Y.
Prove
a. $$\overline{X W} \cong \overline{Z W}$$
b. $$\overline{X V} \cong \overline{Z V}$$
c. ∠VXW ≅ ∠VZW

Maintaining Mathematical Proficiency

Classify the triangle by its sides.

Question 39.

Question 40.

Question 41.

Classify the triangles by its angles.

Question 42.

Question 43.

Question 44.

### 6.2 Bisectors of Triangles

Exploration 1

Properties of the Perpendicular Bisectors of a Triangle

Work with a partner: Use dynamic geometry software. Draw any ∆ABC.
a. Construct the perpendicular bisectors of all three sides of ∆ABC. Then drag the vertices to change ∆ABC. ‘What do you notice about the perpendicular bisectors?

b. Label a point D at the intersection of the perpendicular bisectors.

c. Draw the circle with center D through vertex A of ∆ABC. Then drag the vertices to change ∆ABC. What do you notice?

Exploration 2

Properties of the Angle Bisectors of a Triangle

Work with a partner. Use dynamic geometry software. Draw any ∆ABC.

a. Construct the angle bisectors of all three angles of ∆ABC, Then drag the vertices to change ∆ABC. What do you notice about the angle bisectors?

b. Label a point D at the intersection of the angle bisectors.

c. Find the distance between D and $$\overline{A B}$$. Draw the circle with center D and this distance as a radius. Then drag the vertices to change ∆ABC. What do you notice?
LOOKING FOR STRUCTURE
To be proficient in math, you need to see complicated things as single objects or as being composed of several objects.

Question 3.
What conjectures can you make about the perpendicular bisectors and the angle bisectors of a triangle?

### Lesson 6.2 Bisectors of Triangles

Monitoring Progress

Question 1.
Three snack carts sell hot pretzels horn points A, B, and E. What is the location of the pretzel distributor if it is equidistant from the three carts? Sketch the triangle and show the location.

Find the coordinates of the circumcenter of the triangle with the given vertices.

Question 2.
R(- 2, 5), S(- 6, 5), T(- 2, – 1)

Question 3.
W(- 1, 4), X(1, 4), Y( 1, – 6)

Question 4.
In the figure shown, QM = 3x + 8 and QN = 7x + 2. Find QP.

Question 5.
Draw a sketch to show the location L of the lamppost in Example 4.

### Exercise 6.2 Bisectors of Triangles

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
When three or more lines, rays, or segments intersect in the same Point. they are called _____________ lines, rays, or segments.

Question 2.
WHICH ONE DOESNT BELONG?
Which triangle does not belong with the other three? Explain your reasoning.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4, the perpendicular bisectors of ∆ABC intersect at point G and are shown in blue. Find the indicated measure.

Question 3.
Find BG

Question 4.
Find GA

In Exercises 5 and 6, the angle bisectors of ∆XYZ intersect at point P and are shown in red. Find the indicated measure.

Question 5.
Find PB.

Question 6.
Find HP.

In Exercises 7-10. find the coordinates of the circumcenter of the triangle with the given vertice

Question 7.
A(2, 6), B(8, 6), C(8, 10)

Question 8.
D(- 7, – 1), E(- 1, – 1), F(- 7, – 9)

Question 9.
H(- 10, 7), J(- 6, 3), K(- 2, 3)

Question 10.
L(3, – 6), M(5, – 3) , N (8, – 6)

In Exercises 11-14, N is the incenter of ∆ABC. Use the given information to find the indicated measure.

Question 11.
ND = 6x – 2
NE = 3x + 7
Find NF.

Question 12.
NG = x + 3
NH = 2x – 3
Find NJ.

Question 13.
NK = 2x – 2
NL = – x + 10
Find NM

Question 14.
NQ = 2x
NR = 3x – 2
Find NS.

Question 15.
P is the circumcenter of ∆XYZ. Use the given information to find PZ.
PX = 3x + 2
PY = 4x – 8

Question 16.
P is the circumcenter of ∆XYZ. Use the given information to find PY.
PX = 4x + 3
PZ = 6x – 11

CONSTRUCTION
In Exercises 17-20. draw a triangle of the given type. Find the circumcenter. Then construct the circumscribed circle.

Question 17.
right

Question 18.
obtuse

Question 19.
acute isosceles

Question 20.
equilateral

CONSTRUCTION
In Exercises 21-24, copy the triangle with the given angle measures. Find the incenter. Then construct the inscribed circle.

Question 21.

Question 22.

Question 23.

Question 24.

ERROR ANALYSIS
In Exercises 25 and 26. describe and correct the error in identifying equal distances inside the triangle.

Question 25.

Question 26.

Question 27.
MODELING WITH MATHEMATICS
You and two friends plan to meet to walk your dogs together. You want the meeting place to be the same distance from each person’s house. Explain how you can use the diagram to locate the meeting place.

Question 28.
MODELING WITH MATHEMATICS
You are placing a fountain in a triangular koi pond. YOU want the foutain to be the same distance from each edge of the Pond. Where should you place the fountain? Explain your reasoning. Use a sketch to support your answer.

CRITICAL THINKING
In Exercises 29-32, complete the statement with always, sometimes, or never. Explain your reasoning.

Question 29.
The circumenter of a scalene triangle is ______________ inside the triangle.

Question 30.
If the perpendicular bisector of one side of a triangle intersects the opposite vertex. then the triangle is ______________ isosceles.

Question 31.
The perpendicular bisectors of a triangle intersect at a point that is ______________ equidistant from the midpoints of the sides of the triangle.

Question 32.
The angle bisectors of a triangle intersect at a point that is ______________ equidistant from the sides of the triangle.

CRITICAL THINKING
In Exercises 33 and 34, find the coordinates of the circumcenter of the triangle with the given vertices.

Question 33.
A(2, 5), B(6, 6). C( 12. 3)

Question 34.
D(- 9, – 5), E(- 5, – 9), F(- 2, – 2)

MATHEMATICAL CONNECTIONS
In Exercises 35 and 36. find the a1ue of x that makes N the incenter of the triangle.

Question 35.

Question 36.

Question 37.
PROOF
Where is the circumcenter located in any right triangle? Write a coordinate proof of this result.

Question 38.
PROVING A THEOREM
Write a proof of the Incenter Theorem (Theorem 6.6).
Given ∆ABC, $$\overline{A D}$$ bisects∠CAB,
$$\overline{B D}$$ bisects ∠CBA, $$\overline{D E}$$ ⊥ $$\overline{A B}$$, $$\overline{D F}$$ ⊥ $$\overline{B C}$$, and $$\overline{D G}$$ ⊥ $$\overline{C A}$$
Prove The angle bisectors intersect at D, which is equidistant from $$\overline{A B}$$, $$\overline{B C}$$, and $$\overline{C A}$$

Question 39.
WRITING
Explain the difference between the circumcenter and the incenter of a triangle.

Question 40.
REASONING
Is the incenter of a triangle ever located outside the triangle? Explain your reasoning.

Question 41.
MODELING WITH MATHEMATICS
You are installing a circular pool in the triangular courtyard shown. You want to have the largest pool possible on the site without extending into the walkway.

a. Copy the triangle and show how to install the pool so that it just touches each edge. Then explain how you can he sure that you could not fit a larger pool on the site.
b. You want to have the largest pool possible while leaving at least I foot of space around the pool. Would the center of the pool be in the same position as in part (a)? Justify your answer.

Question 42.
MODELING WITH MATHEMATICS
Archaeologists find three stones. They believe that the stones were once pail of a circle of stones with a community fire pit at its center. They mark the locations of stones A, B, and C on a graph. where distances are measured in feet.

a. Explain how archaeologists can use a sketch to estimate the center of the circle of stones.

b. Copy the diagram and find the approximate coordinates of the point at which the archaeologists should look for the fire pit.

Question 43.
REASONING
Point P is inside ∆ABC and is equidistant from points A and B. On which of the following segments must P be located?
(A) $$\overline{A B}$$
(B) the perpendicular bisector of $$\overline{A B}$$
(C) $$\overline{A C}$$
(D) the perpendicular bisector of $$\overline{A C}$$

Question 44.
CRITICAL THINKING
A high school is being built for the four towns shown on the map. Each town agrees that the school should be an equal distance from each of the tourist towns. Is there a single point where they could agree to build the school? If so, find it. If not, explain why not. Justify your answer with a diagram.

Question 45.
MAKING AN ARGUMENT
Your friend says that the circumcenter of an equilateral triangle is also the incenter of the triangle. Is your friend correct? Explain in your reasoning.

Question 46.
HOW DO YOU SEE IT?
The arms of the windmill are the angle bisectors of the red triangle. What point of concurrency is the point that Connects the three arms?

Question 47.
ABSTRACT REASONING
You are asked to draw a triangle and all its perpendicular bisectors and angle bisectors.
a. For which type of triangle would you need the fewest segments? What is the minimum number of segments you would need? Explain.
b. For which type of triangle would you need the most segments? What is the maximum number of segments you would need? Explain.

Question 48.
THOUGHT PROVOKING
The diagram shows an official hockey rink used by the National Hockey League. Create a triangle using hockey players as vertices in which the center circle is inscribed in the triangle. The center dot should he the incenter of your triangle. Sketch a drawing of the locations of your hockey players. Then label the actual lengths of the sides and the angle measures in your triangle.

COMPARING METHODS
In Exercises 49 and 50. state whether you would use perpendicular bisectors or angle bisectors. Then solve the problem.

Question 49.
You need to cut the largest circle possible from an isosceles triangle made of paper whose sides are 8 inches, 12 inches, and 12 inches. Find the radius of the circle.

Question 50.
On a map of a camp. You need to create a circular walking path that connects the pool at (10, 20), the nature center at (16, 2). and the tennis court at (2, 4). Find the coordinates of the center of the circle and the radius of the circle.

Question 51.
CRITICAL THINKING
Point D is the incenter of ∆ABC. Write an expression for the length x in terms of the three side lengths AB, AC, and BC.

Maintaining Mathematical Proficiency

The endpoints of $$\overline{A B}$$ are given. Find the coordinates of the midpoint M. Then find AB.
Question 52.
A(- 3, 5), B(3, 5)

Question 53.
A(2, – 1), B(10, 7)

Question 54.
A(- 5, 1), B(4, – 5)

Question 55.
A(- 7, 5), B(5, 9)

Write an equation of the line passing through point P that is perpendicular to the given line.
Graph the equations of the lines to check that they are perpendicular.
Question 56.
P(2, 8), y = 2x + 1

Question 57.
P(6, -3), y = – 5

Question 58.
P(- 8, – 6), 2x + 3y = 18

Question 59.
P(- 4, 1), y + 3 = – 4(x + 3)

### 6.3 Medians and Altitudes of Triangles

Exploration 1

Finding Properties of the Medians of a Triangle

Work with a partner. Use dynamic geometry software. Draw any ∆ABC.

a. Plot the midpoint of $$\overline{B C}$$ and label it D, Draw $$\overline{A D}$$, which is a median of ABC. Construct the medians to the other two sides of ∆ABC.

b. What do you notice about the medians? Drag the vertices to change ∆ABC. Use your observations to write a conjecture about the medians of a triangle.

c. In the figure above, point G divides each median into a shorter segment and a longer segment. Find the ratio of the length of each longer segment to the length of the whole median. Is this ratio always the same? Justify your answer.

Exploration 2

Finding Properties of the Altitudes of a Triangle

Work with a partner. Use dynamic geometry software. Draw any ∆ABC.

a. Construct the perpendicular segment from vertex A to $$\overline{B C}$$. Label the endpoint D. $$\overline{A D}$$ is an altitude of ∆ABC.

b. Construct the altitudes to the other two sides of ∆ABC. What do you notice?

c. Write a conjecture about the altitudes of a triangle. Test your conjecture by dragging the vertices to change ∆ABC.
LOOKING FOR STRUCTURE
To be proficient in math, you need to look closely to discern a pattern or structure.

Question 3.
What conjectures can you make about the medians and altitudes of a triangle?

Question 4.
The length of median $$\overline{R U}$$ in ∆RST is 3 inches. The point 0f concurrency of the three medians of ∆RST divides $$\overline{R U}$$ into two segments. What are the lengths of these two segments?

### Lesson 6.3 Medians and Altitudes of Triangles

Monitoring Progress

There are three paths through a triangular park. Each path goes from the midpoint of one edge to the opposite corner. The paths meet at point P.

Question 1.
Find PS and PC when SC = 2100 feet.

Question 2.
Find TC and BC when BT = 10oo feet.

Question 3.
Find PA and TA when PT = 800 feet.

Find the coordinates of the centroid of the triangle with the given vertices.

Question 4.
F(2, 5), G(4, 9), H(6, 1)

Question 5.
X(- 3, 3), Y(1, 5), Z(- 1, – 2)

Tell whether the orthocenter of the triangle with the given vertices is inside, on, or outside the triangle. Then find the coordinates of the orthocenter.

Question 6.
A(0, 3), B(0, – 2), C(6, -,3)

Question 7.
J(- 3, – 4), K(- 3, 4), L(5, 4)

Question 8.
WHAT IF?
In Example 4, you want to show that median $$\overline{B D}$$ is also an angle bisector. How would your proof be different?

### Exercise 6.3 Medians and Altitudes of Triangles

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Name the four types of points of concurrency. Which lines intersect to form each of the points?

Question 2.
COMPLETE THE SENTENCE
The length of a segment from a vertex to the centroid is ______________ the length of the median from that vertex.

Monitoring progress and Modeling with Mathematics

In Exercises 3-6, point P is the centroid of ∆LMN. Find PN and QP.

Question 3.
QN = 9

Question 4.
QN = 21

Question 5.
QN = 30

Question 6.
QN = 42

In Exercises 7-10. point D is the centroid of ∆ ABC. Find CD and CE.

Question 7.
DE = 5

Question 8.
DE = 11

Question 9.
DE = 9

Question 10.
DE = 15

In Exercises 11-14. point G is the centroid of ∆ABC. BG = 6, AF = 12, and AE = 15. Find the length of the segment.

Question 11.
$$\overline{F C}$$

Question 12.
$$\overline{B F}$$

Question 13.
$$\overline{A G}$$

Question 14.
$$\overline{G E}$$

In Exercises 15-18. find the coordinates of the centroid of the triangle with the given vertices.

Question 15.
A(2, 3), B(8, 1), C(5, 7)

Question 16.
F(1, 5), G( – 2, 7), H(- 6, 3)

Question 17.
S(5, 5), T(11, – 3), U(- 1, I)

Question 18.
X(1, 4), Y(7, 2), Z(2, 3)

In Exercises 19-22. tell whether the orthocenter is inside, on, or outside the triangle. Then find the coordinates of the orthocenter.

Question 19.
L(0, 5), M(3, 1), N(8, 1)

Question 20.
X(- 3, 2), Y(5, 2), Z(- 3, 6)

Question 21.
A(- 4, 0), B(1, 0), C(- 1, 3)

Question 22.
T(-2, 1), U( 2, 1), V(0, 4)

CONSTRUCTION
In Exercises 23-26, draw the indicated triangle and find its centroid and orthocenter.
Question 23.
isosceles right triangle

Question 24.
obtuse scalene triangle

Question 25.
right scalene triangle

Question 26.
acute isosceles triangle

ERROR ANALYSIS
In Exercises 27 and 28, describe and correct the error in finding DE. Point D is the centroid of ∆ABC.

Question 27.

Question 28.

PROOF
In Exercises 29 and 30, write a proof of the statement.

Question 29.
The angle bisector from the vertex angle to the base of an isosceles triangle is also a median.

Question 30.
The altitude from the vertex angle to the base of an isosceles triangle is also a perpendicular bisector.

CRITICAL THINKING
In Exercises 31-36, complete the statement with always, sometimes, or never. Explain your reasoning.

Question 31.
The centroid is _____________ on the triangle.

Question 32.
The orthocenter is _____________ outside the triangle.

Question 33.
A median is _____________ the same line segment as a perpendicular bisector.

Question 34.
An altitude is ______________ the same line segment as an angle bisector.

Question 35.
The centroid and orthocenter are _____________ the same point.

Question 36.
The centroid is ______________ formed by the intersection oÍ the three medians.

Question 37.
WRITING
Compare an altitude of a triangle with a perpendicular bisector of a triangle.

Question 38.
WRITING
Compare a median. an altitude, and an angle bisector of a triangle.

Question 39.
MODELING WITH MATHEMATICS
Find the area of the triangular part of the paper airplane wing that is outlined in red. Which special segment of the triangle did you use?

Question 40.
ANALYZING RELATIONSHIPS
Copy and complete the statement for ∆DEF with centroid K and medians
$$\overline{D H}$$, $$\overline{E J}$$, and $$\overline{F G}$$.

a. EJ = ____ KJ

b. DK = ____ KH

c. FG = ___ KF

d. KG = ___ FG

MATHEMATCAL CONNETIONS
In Exercises 41-44, point D is the centroid of ∆ABC. Use the given information to find the value of x.

Question 41.
BD = 4x + 5 and BF = 9x

Question 42.
GD = 2x – 8 and GC = 3x + 3

Question 43.
AD = 5x and DE = 3x – 2

Question 44.
DF = 4x – 1 and BD = 6x + 4

Question 45.
MATHEMATICAL CONNECTIONS
Graph the lines on the same coordinate plane. Find the centroid of the triangle formed by their intersections.
y1 = 3x – 4
y2 = $$\frac{3}{4}$$x + 5
y2 = – $$\frac{3}{2}$$x – 4

Question 46.
CRITICAL THINKING
In what types of triangles can a vertex be one of the points of concurrency of the triangle? Explain your reasoning.

Question 47.
WRITING EQUATIONS
Use the numbers and symbols to write three different equations for PE.

Question 48.
HOW DO YOU SEE IT?
Use the figure.

a. What type of segment is $$\overline{K M}$$? Which point of concurrency lies on $$\overline{K M}$$?

b. What type of segments is $$\overline{K N}$$? Which point of concurrency lies on $$\overline{K N}$$?

c. Compare the areas of ∆JKM and ∆KLM. Do you think the areas of the triangles formed by the median of any triangle will always compare this way? Explain your reasoning.

Question 49.
MAKING AN ARGUMENT
Your friend claims that it is possible for the circumcenter, incenter, centroid, and orthocenter to all be the same point. Do you agree? Explain your reasoning.

Question 50.
DRAWING CONCLUSIONS
The center of gravity of
a triangle, the point where a triangle can balance on the tip of a pencil, is one of the four points of concurrency. Draw and cut out a large scalene triangle on a piece of cardboard. Which of the four points of concurrency is the center of gravity? Explain.

Question 51.
PROOF
Prose that a median of an equilateral triangle is also an angle bisector, perpendicular bisector, and altitude.

Question 52.
THOUGHT PROVOKING
Construct an acute scalene triangle. Find the orthocenter, centroid, and circumcenter. What can you conclude about the three points of concurrency?

Question 53.
CONSTRUCTION
Follow the steps to construct a nine-point circle. Why is it called a nine-point circle?
Step 1 Construct a large acute scalene triangle.
Step 2 Find the orthocenter and circumcenter of the triangle.
Step 3 Find the midpoint between the orthocenter and circumcenter.
Step 4 Find the midpoint between each vertex and the orthocenter.
Step 5 Construct a circle. Use the midpoint in Step 3 as the center of the circle, and the distance from the center to the midpoint of a side of the triangle as the radius.

Question 54.
PROOF
Prove the statements in parts (a)-(c).
Given $$\overline{L P}$$ and $$\overline{M Q}$$ are medians of scalenc ∆LMN.
Point R is on $$\vec{L}$$P such that $$\overline{L P} \cong \overline{P R}$$. Point S is on $$\vec{M}$$Q such that $$\overline{M Q} \cong \overline{Q S}$$.
Prove
a. $$\overline{N S} \cong \overline{N R}$$
b. $$\overline{N S}$$ and $$\overline{N R}$$ are both parallel to $$\overline{L M}$$.
c. R, N, and S are collinear.

Maintaining Mathematical Proficiency

Determine whether $$\overline{A B}$$ is parallel to $$\overline{C D}$$.

Question 55.
A(5, 6), B (- 1, 3), C(- 4, 9), D(- 16, 3)

Question 56.
A(- 3, 6), B(5, 4), C(- 14, – 10), D(- 2, – 7)

Question 57.
A (6, – 3), B(5, 2), C(- 4, – 4), D(- 5, 2)

Question 58.
A(- 5, 6), B(- 7, 2), C(7, 1), D(4, – 5)

### 6.1 and 6.3 Quiz

Find the indicated measure. Explain your reasoning.

Question 1.
UV

Question 2.
QP

Question 3.
m∠GJK

Find the coordinates of the circumcenter of the triangle with the given vertices.

Question 4.
1(- 4, 2), B(- 4, – 4), C(0, – 4)

Question 5.
D(3, 5), E(7, 9), F(11, 5)

The incenter of ∆ABC is point N. Use the given information to find the indicated measure.

Question 6.
NQ = 2x + 1, NR = 4x – 9
Find NS.

Question 7.
NU = – 3x + 6, NV = – 5x
Find NT.

Question 8.
NZ = 4x – 10, NY = 3x – 1
Find NW.

Find the coordinates of the centroid of the triangle wilt the given vertices.
Question 9.
J(- 1, 2), K(5, 6), L(5, – 2)

Question 10.
M(- 8, – 6), N(- 4, – 2), P(0, – 4)

Tell whether the orthocenter is inside, on, or outside the triangle. Then find its coordinates.

Question 11.
T(- 2, 5), U(0, 1), V(2, 5)

Question 12.
X(- 1, – 4), Y(7, – 4), Z(7, 4)

Question 13.
A woodworker is culling the largest wheel possible from a triangular scrap of wood. The wheel just touches each side of the triangle, as shown.

a. Which point of concurrency is the center of the circle? What type
of segments are $$\overline{B G}$$, $$\overline{C G}$$, and $$\overline{A G}$$?

b.
Which theorem can you use to prove that ∆BGF ≅ ∆BGE?

c. Find the radius of the wheel to the nearest tenth of a centimeter. Justify your answer.

Question 14.
The Deer County Parks Committee plans to build a park at point P, equidistant from the three largest cities labeled X, Y, and Z. The map shown was created b the committee.

a. Which point of concurrency did the commIttee use as the location of the Park?

b. Did the committee use the best point of concurrency for the location of the park? Ii not, which point would be better to use? Explain.

### 6.4 The Triangle Midsegment Theorem

Exploration 1

Midsegments of a Triangle

Work with a partner. Use dynamic geometry software. Draw any ∆ABC.

a. Plot midpoint D of $$\overline{A B}$$ and midpoint E of $$\overline{B C}$$. Draw $$\overline{D E}$$, which is a midsegment of ∆ABC.

b. Compare the slope and length of $$\overline{D E}$$ with the slope and length of $$\overline{A C}$$.

c. Write a conjecture about the relationships between the midsegments and sides of a triangle. Test your conjecture by drawing the other midsegments of ∆ABC, dragging vertices to change ∆ABC. and noting whether the relationships hold.

Exploration 2

Midsegments of a Triangle

Work with a partner. Use dynamic geometry software. Draw any ∆ABC.

a. Draw all three midsegments of ∆ABC.

b. Use the drawing to write a Conjecture about the triangle formed by the midsegments of the original triangle.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

Question 3.
How are the midsegments of a triangle related to the sides of the triangle?

Question 4.
In ∆RST. $$\overline{U V}$$ is the rnidsegrnent connecting the midpoints of $$\overline{R S}$$ and $$\overline{S T}$$. Given
UV = 12, find RT.

### Lesson 6.4 The Triangle Midsegment Theorem

Monitoring progress

Use the graph of △ABC.

Question 1.
In △ABC, show that midsegments $$\overline{D E}$$ is parallel to $$\overline{A C}$$ and that DE = $$\frac { 1 }{ 2 }$$AC.

Question 2.
Find the coordinates of the endpoints of midsegments $$\overline{E F}$$, Which opposite $$\overline{A B}$$. show that $$\overline{E F}$$ is parallel to $$\overline{A B}$$ and that EF = $$\frac { 1 }{ 2 }$$AB.

Question 3.
In Example 2, find the coordinates of F, the midpoint of $$\overline{O C}$$. Show that $$\overline{F E}$$ || $$\overline{O B}$$ and FE = $$\frac { 1 }{ 2 }$$OB.

Question 4.
Copy the diagram in Example 3. Draw and name the third midsegment.
Then find the length of $$\overline{V S}$$ when the length of the third midsegment is 81 inches.

Question 5.
In Example 4. if F is the midpoint of $$\overline{C B}$$, what do you know about $$\overline{D F}$$?

Question 6.
WHAT IF?
In Example 5, you jog down Peach Street to Plum Street, over Plum Street to Cherry Street. up Cherry Street to Pear Street. over Pear Street to Peach Street. and then back home up Peach Street. Do you jog more miles in Example 5? Explain.

### Exercise 6.4 The Triangle Midsegment Theorem

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
The ___________ of a triangle is a segment that connects the midpoints of two sides of the triangle.

Question 2.
COMPLETE THE SENTENCE
If $$\overline{D E}$$ is the midsegment opposile $$\overline{A C}$$ in ∆ABC, then $$\overline{D E}$$ || $$\overline{A C}$$ and DE = ________ AC by the Triangle Midsegrnent Theorem (Theorem 6.8).

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, use the graph of ∆ABC with midsegments $$\overline{D E}$$, $$\overline{E F}$$, and $$\overline{D F}$$.

Question 3.
Find the coordinates of points D, E, and F.

Question 4.
Show that $$\overline{D E}$$ is parallel to $$\overline{C B}$$ and that DE = $$\frac{1}{2}$$CB.

Question 5.
Show that $$\overline{E F}$$ is parallel to $$\overline{A C}$$ and that EF = $$\frac{1}{2}$$AC.

Question 6.
Show that $$\overline{D F}$$ is parallel to $$\overline{A B}$$ and that DF = $$\frac{1}{2}$$AB.

In Exercises 7-10, $$\overline{D E}$$ is a midsegment of ∆ABC Find the value of x.

Question 7.

Question 8.

Question 9.

Question 10.

In Exercise 11-16, $$\overline{X J} \cong \overline{J Y}$$, $$\overline{Y L} \cong \overline{L Z}$$, and $$\overline{X K} \cong \overline{K Z}$$. Copy and complete the statement.

Question 11.
$$\overline{J K}$$ || __________

Question 12.
$$\overline{X Y}$$ || __________

Question 13.
$$\overline{J L}$$ || __________

Question 14.
$$\overline{J L}$$ ≅ __________ ≅ __________

Question 15.
$$\overline{J Y}$$ ≅ __________ ≅ __________

Question 16.
$$\overline{J K}$$ ≅ __________ ≅ __________

MATHEMATICAL CONNECTIONS
In Exercises 17-19. use ∆GHJ, where A, B, and C are midpoints of the sides.

Question 17.
When AB = 3x + 8 and GJ = 2x + 24, what is AB?

Question 18.
When AC = 3y – 5 and HJ = 4y + 2, what is HB?

Question 19.
When GH = 7 – 1 and CB = 4z – 3. what is GA?

Question 20.
ERROR ANALYSIS
Describe and correct the error.

Question 21.
MODELING WITH MATHEMATICS
The distance between consecutive bases on a baseball held is 90 feet. A second baseman stands halfway between first base and second base, a shortstop stands hallway between second base and third base, and a pitcher stands halfway between first base and third base. Find the distance between the shortstop and the pitcher.

Question 22.
PROVING A THEOREM
Use the figure from Example 2 to prose the Triangle Midsegment Theorem (Theorem 6.8) for midsegment $$\overline{D F}$$, where F is the midpoint of $$\overline{O C}$$.

Question 23.
CRITICAL THINKING
$$\overline{X Y}$$ is a midsegment of ∆LMN. Suppose $$\overline{D E}$$ is called a “quarter segment” of ∆LMN. What do you think an “eighth segment” would be? Make conjectures about the properties of a quarter segment and an eighth segment. Use variable coordinates to verify your conjectures.

Question 24.
THOUGHT PROVOKING
Find a real-life object that uses midsegments as part of its structure. Print a photograph of the object and identify the midsegments of one of the triangles in the structure.

Question 25.
ABSTRACT REASONING
To create the design shown. shade the triangle formed by the three midsegments of the triangle. Then repeat the process for each unshaded triangle.

a. What is the perimeter of the shaded triangle in Stage 1?
b. What is the total perimeter of all the shaded triangles in Stage 2?
c. What is the total perimeter of all the shaded triangles in Stage 3?

Question 26.
HOW DO YOU SEE IT?
Explain how you know that the yellow triangle is the midsegment triangle of the red triangle in the pattern of floor tiles shown.

Question 27.
ATTENDING TO PRECISION
The points P(2, 1), Q(4, 5), and R(7, 4) are the midpoints of the sides of a triangle. Graph the three midsegments. Then show how to use your graph and the properties of midsemeriLs to draw the original triangle. Give the coordinates of each vertex.

Maintaining Mathematical Proficiency

Find a counter example to show that the conjecture is false.

Question 28.
The difference of two numbers is always less than the greater number.

Question 29.
An isosceles triangle is always equilateral.

### 6.5 Indirect Proof and Inequalities in One Triangle

Exploration 1

Comparing Angle Measures and Side Lengths

Work with a partner: Use dynamic geometry software. Draw any scalene ∆ABC

a. Find the side lengths and angle measures of the triangle.

b. Order the side lengths. Order the angle measures. What do you observe?

c. Drug the vertices of ∆ABC to form new triangles. Record the side lengths and angle measures in a table. Write a conjecture about your findings.

Exploration 2

A Relationship of the Side Lengths of a Triangle

Work with a partner. Use dynamic geometry software. Draw any ∆ABC.

a. Find the side lengths of the triangle.

b. Compare each side length with the sum of the other two side lengths.
ATTENDING TO PRECISION
To be proficient in math, you need to express numerical answers with a degree of precision appropriate for the content.

c. Drag the vertices of ∆ABC to form new triangles and repeat parts (a) and (b). Organize your results in a table. Write a conjecture about your findings.

Question 3.
How are the sides related to the angles of a triangle? How are any two sides of a triangle related to the third side?

Question 4.
Is it possible for a triangle to have side lengths of 3, 4, and 10? Explain.

### Lesson 6.5 Indirect Proof and Inequalities in One Triangle

Monitoring Progress

Question 1.
Write in indirect proof that a scalene triangle cannot have two congruent angles.

Question 2.
List the angle of ∆PQR in order from smallest to largest.

Question 3.
List the sides of ∆RST in order from shortest to longest.

Question 4.
A triangle has one side of length 12 inches and another side of length 20 inches. Describe the possible lengths of the third side.

Decide Whether it is possible to construct a triangle with the given side lengths. Explain your reasoning.

Question 5.
4 ft, 9 ft, 10 ft

Question 6.
8 ft, 9 ft, 18 ft

Question 7.
5 cm, 7 cm, 12 cm

### Exercise 6.5 Indirect Proof and Inequalities in One Triangle

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Why is an indirect proof also called a proof by contradiction?

Question 2.
WRITING
How can you tell which side of a triangle is the longest Irom the angle measures of the
triangle? How can you tell which side is the shortest?

Monitoring progress and Modeling with Mathematics

In Exercises 3-6, write the first step in an indirect proof of the statement.

Question 3.
If WV + VU ≠ 12 inches and VU = 5 inches, then WV ≠ 7 inches.

Question 4.
If x and y are odd integers. then xy is odd.

Question 5.
In ∆ABC. if m∠A = 100°, then ∠B is not a right angle.

Question 6.
In ∆JKL, if M is the midpoint of $$\overline{K L}$$, then $$\overline{J M}$$ is a median.

In Exercises 7 and 8, determine which two statements contradict each other. Explain your reasoning.

Question 7.
(A) ∆LMN is a right triangle.
(B) ∠L ≅∠V
(C) ∆LMN is equilateral.

Question 8.
(A) Both ∠X and ∠Y have measures greater than 20°.
(B) Both ∠X and ∠Y have measures less than 30°.
(C) m∠X + m∠Y = 62°

In Exercises 9 and 10, use a ruler and protractor to draw the given type of triangle. Mark the largest angle and longest side in red and the smallest angle and shortest side in blue. What do you notice?

Question 9.
acute scalene

Question 10.
right scalene

In Exercises 11 and 12, list the angles of the given triangle from smallest to largest.

Question 11.

Question 12.

In Exercises 13-16, list the sides of the given triangle from shortest to longest.

Question 13.

Question 14.

Question 15.

Question 16.

In Exercises 17-20, describe the possible lengths of the third side of the triangle given the lengths of the other to sides.

Question 17.
5 inches, 12 inches

Question 18.
12 feet, 18 feet

Question 19.
2 feet, 40 inches

Question 20.
25 meters, 25 meters

In Exercises 21-23, is it possible to construct a triangle with the given side lengths? If not, explain why not.

Question 21.
6, 7, 11

Question 22.
3, 69

Question 23.
28, 17, 46

Question 24.
35, 120, 125

Question 25.
ERROR ANALYSIS
Describe and correct the error in writing the first step of an indirect proof.

Question 26.
ERROR ANALYSIS
Describe and correct the error in labeling the side lengths 1, 2, and √3 on the triangle.

Question 27.
REASONING
You are a lawyer representing a client who has been accused of a crime. The crime took place in Los Angeles, California. Security footage shows your client in New York at the time of the crime. Explain how to use indirect reasoning to prove your client is innocent.

Question 28.
REASONING
Your class has fewer than 30 students. The teacher divides your class into two groups. The first group has 15 students. Use indirect reasoning to show that the second group must have fewer than 15 students.

Question 29.
PROBLEM SOLVING
Which statement about ∆TUV is false?

(A) UV > TU
(B) UV + TV > TU
(C) UV < TV
(D) ∆TUV is isosceles.

Question 30.
PROBLEM SOLVING
In ∆RST. which is a possible side length for ST? Select all that apply.

(A) 7
(B) 8
(C) 9
(D) 10

Question 31.
PROOF
Write an indirect proof that an odd number is not divisible by 4.

Question 32.
PROOF
Write an indirect proof of the statement
“In ∆QRS, if m∠Q + m∠R = 90°, then m∠S = 90°.”

Question 33.
WRITING
Explain why the hypotenuse of a right triangle must always be longer than either leg.

Question 34.
CRITICAL THINKING
Is it possible to decide if three side lengths form a triangle without checking all three inequalities shown in the Triangle Inequality Theorem (Theorem 6. 11)? Explain your reasoning.

Question 35.
MODELING WITH MATHEMATICS
You can estimate the width of the river from point A to the tree at point B by measuring the angle to the tree at several locations along the riverbank. The diagram shows the results for locations C and D.

a. Using ∆BCA and ∆BDA, determine the possible widths of the river. Explain your reasoning.
b. What could you do if you wanted a closer estimate?

Question 36.
MODELING WITH MATHEMATICS
You travel from Fort Peck Lake to Glacier National Park and from Glacier National Park to Granite Peak.

a. Write two inequalities to represent the possible distances from Granite Peak back to Fort Peck Lake.

b. How is your answer to part (a) affected if you know that m∠2 < m∠1 and m∠2 < m∠3?

Question 37.
REASONING
In the figure. $$\overline{X Y}$$ bisects ∠WYZ. List all six angles of ∆XYZ and ∆WXY in order from smallest to largest. Explain ‘our reasoning.

Question 38.
MATHEMATICAL CONNECTIONS
In ∆DEF, m∠D = (x + 25)°. m∠E = (2x – 4)°, and in m∠F = 63°. List the side lengths and angle measures of the triangle in order from least to greatest.

Question 39.
ANALYZING RELATIONSHIPS
Another triangle inequality relationship is given by the Exterior Angle Inequality Theorem. It states:

The measures of an exterior angle of a triangle is greater than the measure of either of the nonadjacent interior angles.

Explain how you know that m∠1 > m∠A and m∠1 > m∠B in ∆ABC with exterior angle ∠1.

MATHEMATICAL CONNECTIONS
In Exercises 40 and 41, describe the possible values of x.

Question 40.

Question 41.

Question 42.
HOW DO YOU SEE IT?
Your house is on the corner of Hill Street and Eighth Street. The library is on the corner of View Street and Seventh Street. What is the shortest route to et from your house to the library? Explain your reasoning.

Question 43.
PROVING A THEOREM
Use the diagram to prove the Triangle Longer Side Theorem (Theorem 6.9).

Given BC > AB, BD = BA
Prove m∠BAC > m∠C

Question 44.
USING STRUCTURE
The length of the base of an isosceles triangle is l. Describe the possible lengths for each leg. Explain our reasoning.

Question 45.
MAKING AN ARGUMENT
Your classmate claims to have drawn a triangle with one side length of 13 inches and a perimeter of 2 feet. Is this possible? Explain your reasoning.

Question 46.
THOUGHT PROVOKING
Cut two pieces of string that are each 24 centimeters long. Construct an isosceles triangle out of one string and a scalene triangle out of the other. Measure and record the side lengths. Then classify each triangle by its angles.

Question 47.
PROVING A THEOREM
Prove the Triangle Inequality Theorem (Theorem 6. 11).
Given ∆ABC
Prove AB + BC > AC, AC + BC > AB, and AB + AC > BC

Question 48.
ATTENDING TO PRECISION
The perimeter of ∆HGF must be between what two integers? Explain your reasoning.

Question 49.
PROOF
Write an indirect proof that a perpendicular segment is the shortest segment from a point to a plane.

Given $$\overline{P C}$$ ⊥ palne M
Prove $$\overline{P C}$$ is the shortest from P to plane M.

Maintaining Mathematical Proficiency

Name the indicated angle between the pair of sides given.

Question 50.
$$\overline{A E}$$ and $$\overline{B E}$$

Question 51.
$$\overline{A C}$$ and $$\overline{D C}$$

Question 52.
$$\overline{A C}$$ and $$\overline{D C}$$

Question 53.
$$\overline{C E}$$ and $$\overline{B E}$$

### 6.6 Inequalities in Two Triangles

Exploration 1

Comparing Measures in Triangles

Work with a partner. Use dynamic geometry software.

a. Draw ∆ABC, as shown below.

b. Draw the circle with center C(3, 3) through the point A(1, 3).

c. Draw ∆DBC so that D is a point on the circle.

d. Which two sides of ∆ABC are congruent to two sides of ∆DBC? Justify your answer.

e. Compare the lengths of $$\overline{A B}$$ and $$\overline{D B}$$. Then compare the measures of ∠ACB and ∠DCB. Are the results what you expected? Explain.

f. Drag point D to several locations on the circle. At each location, repeat part (e). Copy and record your results in the table below.

g. Look for a pattern of the measures in our table. Then write a conjecture that summarizes your observations.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

Question 2.
If two sides of one triangle are congruent to two sides of another triangle, what can you say about the third sides of the triangles?

Question 3.
Explain how you can use the hinge shown at the left to model the concept described in Question 2.

### Lesson 6.6 Inequalities in Two Triangles

Monitoring Progress

Use the diagram

Question 1.
If PR = PS and m∠QPR > m∠QPS, which is longer, $$\overline{S Q}$$ or $$\overline{R Q}$$?

Question 2.
If PR = PS and RQ < SQ, which is larger, ∠RPQ or ∠SPQ?

Question 3.
Write a temporary assumption you can make to prove the Hinge Theorem indirectly. What two cases does that assumption lead to?

Question 4.
WHAT IF?
In Example 5, Group C leaves camp and travels 2 miles due north. then turns 40° towards east and travels 1.2 miles. Compare the distances from camp for all three groups.

### Exercise 6.6 Inequalities in Two Triangles

Vocabulary and Core Concept Check

Question 1.
WRITING
Explain why Theorem 6.12 is named the “Hinge Theorem.”

Question 2.
COMPLETE THE SENTENCE
In ∆ABC and ∆DEF, $$\overline{A B} \cong \overline{D E}$$, $$\overline{B C} \cong \overline{E F}$$. and AC < DF. So m∠_______ > m ∠ _______ by the Converse of the Hinge Theorem

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, copy and complete the statement with <, >, or = Explain your reasoning.

Question 3.
m∠1 ________ m∠2

Question 4.
m∠1 ________ m∠2

Question 5.
m∠1 ________ m∠2

Question 6.
m∠1 ________ m∠2

In Exercises 7-10. copy and complete the statement with <, >, or =. Explain your reasoning.

Question 7.

Question 8.
MN ________ LK

Question 9.
TR ________ UR

Question 10.
AC ________ DC

PROOF
In Exercises 11 and 12, write a proof.

Question 11.
Given $$\overline{X Y} \cong \overline{Y Z}$$, m∠WYZ > m∠WYX
Prove WZ > WX

Question 12.
Given $$\overline{B C} \cong \overline{D A}$$, DC < AB
Prove m∠BCA > m∠DAC

In Exercises 13 and 14, you and your friend leave on different flights from the same airport. Determine which flight is farther from the airport. Explain your reasoning.

Question 13.
Your flight: Flies 100 miles due west, then turns 20° toward north and flies 50 miles.
Friend’s flight: Flies 100 miles due north, then turns 30° toward east and flies 50 miles.

Question 14.
Your flight: Flies 21omiles due south, then turns 70° toward west and flies 80 miles.
Friend’s flight: Flies 80 miles due north, then turns 50° toward east and flies 210 miles.

Question 15.
ERROR ANALYSIS
Describe and correct the error in using the Hinge Theorem (Theorem 6.12).

Question 16.
REPEATED REASONING
Which is possible measure for ∠JKM? Select all that apply.

(A) 15°
(B) 22°
(C) 25°
(D) 35°

Question 17.
DRAWING CONCLUSIONS
The path from E to F is longer than the path from E to D. The path from G to D is the same length as the path from G to F. What can you conclude about the angles of the paths? Explain your reasoning.

Question 18.
ABSTRACT REASONING
In ∆EFG, the bisector of ∠F intersects the bisector of ∠G at point H. Explain why $$\overline{F G}$$ must be longer than $$\overline{F H}$$ or $$\overline{H G}$$.

Question 19.
ABSTRACT REASONING
$$\overline{N R}$$ is a median of ∆NPQ, and NQ > NP Explain why ∠NRQ is obtuse.

MATHEMATICAL CONNECTIONS
In Exercises 20 and 21, write and solve an inequality for the possible values of x.

Question 20.

Question 21.

Question 22.
HOW DO YOU SEE IT?
In the diagram, triangles are formed by the locations of the players on the basketball court. The dashed lines represent the possible paths of the basketball as the players pass. How does m∠ACB compare with m∠ACD?

Question 23.
CRITICAL THINKING
In ∆ABC, the altitudes from B and C meet at point D, and m∠BDC. What is true about ∆ABC? Justify your answer.

Question 24.
THOUGHT PROVOKING
The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, state an inequality involving the sum of the angles of a triangle. Find a formula for the area of a triangle in spherical geometry.

Maintaining Mathematical proficiency

Find the value of x.

Question 25.

Question 26.

Question 27.

Question 28.

### 6.1 Perpendicular and Angle Bisectors

Find the indicated measure. Explain your reasoning.

Question 1.
DC

Question 2.
RS

Question 3.
m∠JFH

### 6.2 Bisectors of Triangles

Find the coordinates of the circumcenter of the triangle with the given vertices.

Question 4.
T(- 6, – 5), U(0, – 1), V(0, – 5)

Question 5.
X(- 2, 1), Y(2, – 3), Z(6, – 3)

Question 6.
Point D is the incenter of ∆LMN. Find the value of x.

### 6.3 Medians and Altitudes of Triangles

Find the coordinates of the centroid of the triangle with the given vertices.

Question 7.
A(- 10, 3), B(- 4, 5), C(- 4, 1)

Question 8.
D(2, – 8), E(2, – 2), F(8, – 2)

Tell whether the orthocenter of the triangle with the given vertices is inside, on, or outside the triangle. Then find the coordinates of the orthocenter.

Question 9.
G(1, 6), H(5, 6), J(3, 1)

Question 10.
K(-8, 5), L(- 6, 3), M(0, 5)

### 6.4 The Triangle Midsegment Theorem

Find the coordinates of the vertices of the midsegment triangle for the triangle with the given vertices.

Question 11.
A(- 6, 8), B(- 6, 4), C(0, 4)

Question 12.
D(- 3, 1), E(3, 5), F(1, – 5)

### 6.5 Indirect Proof and Inequalities in One Triangle

Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.

Question 13.
4 inches, 8 inches

Question 14.
6 meters, 9 meters

Question 15.
11 feet, 18 feet

Question 16.
Write an indirect proof 0f the statement “In ∆XYZ, if XY = 4 and XZ = 8. then YZ > 4.”

### 6.6 Inequalities in Two Triangles

Use the diagram.

Question 17.
If RQ = RS and m∠QRT > m∠SRT, then how does $$\overline{Q T}$$ Compare to $$\overline{S T}$$?

Question 18.
If RQ = RS and QT > ST, then how does ∠QRT compare to ∠SRT?

### Relationships Within Triangles Chapter Test

In Exercise 1 and 2, $$\overline{M N}$$ is a midsegment of ∆JKL. Find the value of x.

Question 1.

Question 2.

Find the indicated measure. Identify the theorem you use.

Question 3.
ST

Question 4.
WY

Question 5.
BW

Copy and complete the statement with <, >, or =.

Question 6.
AB _____ CB

Question 7.
m∠1 _____ m∠2

Question 8.
m∠MNP ________ m∠NPM

Question 9.
Find the coordinates of the circumcenter, orthocenter, and centroid of the triangle with vertices A(0, – 2), B(4, – 2), and C(0, 6).

Question 10.
Write an indirect proof of the Corollary to the Base Angles Theorem (Corollary 5.2): If ∆PQR is equilateral, then it is equiangular.

Question 11.
∆DEF is a right triangle with area A. Use the area for ∆DEF to write an expression for the area of ∆GEH. Justify your answer.

Question 12.
Two hikers start at a visitor center. The first hikes 4 miles hikes due west, then turns 40° toward south and hikes 1.8 miles. The second hikes 4 miles due east, then turns 52° toward north and hikes 1 .8 miles. Which hiker is farther from the visitor center? Explain how you know.

In Exercises 13-15, use the map.

Question 13.
Describe the possible lengths of Pine Avenue.

Question 14.
You ride your bike along a trail that represents the shortest distance from the beach to Main Street. You end up exactly halfway between your house and the movie theatre. How long is Pine Avenue? Explain.

Question 15.
A market is the same distance from your house, the movie theater, and the beach. Copy the map and locate the market.

### Relationships Within Triangles Cumulative Assessment

Question 1.
Which definitions(s) and/or theorem(s) do you need to use to prove the Converse of the Perpendicular Bisector Theorem (Theorem 6.2)? Select all that apply.
Given CA = CB
Prove Point C lies on the perpendicular bisector of AB.

 Definition of perpendicular bisector Definition of angle bisector Definition of segment congruence Definition of angle congruence Base Angles Theorem (Theorem 5.6) Converse of the Base Angles Theorem (Theorem 5.7) ASA Congruence Theorem (Theorem 5.10) AAS Congruence Theorem (Theorem 5.11)

Question 2.
Use the given information to write a two-column proof.
Given $$\overline{Y G}$$ is the perpendicular bisector of $$\overline{D F}$$.
Prove ∆DEY ≅ ∆FEY

Question 3.
What are the coordinates of the centroid of ∆LMN?

(A) (2, 5)
(B) (3, 5
(C) (4, 5
(D) (5, 5)

Question 4.
Use the steps in the construction to explain how you know that the circle is circumscribed about ∆ABC.

Question 5.
Enter the missing reasons in the proof of the Base Angles Theorem (Theorem 5.6).

Given $$\overline{A B} \cong \overline{\Lambda C}$$
Prove ∠B = ∠C

 Statements Reasons 1. Draw $$\overline{A D}$$, the angle bisector of ∠CAB 1. Construction of angle bisector 2. ∠CAD ≅ ∠BAD 2. ________________________ 3. $$\overline{\Lambda B} \cong \overline{A C}$$ 3. ________________________ 4. $$\overline{D A} \cong \overline{D A}$$ 4. ________________________ 5. ∆ADB ≅ ∆ADC 5. ________________________ 6. ∠B ≅ ∠C 6. ________________________

Question 6.
Use the graph of ∆QRS.

a. Find the coordinates of the vertices of the midsegment triangle. Label the vertices T, U, and V.

b. Show that each midsegment joining the midpoints of two sides is parallel to the third side and is equal to half the length of the third side.