# Spectrum Math Grade 8 Chapter 5 Lesson 5 Answer Key Slope and Similar Triangles

Students can use the Spectrum Math Grade 8 Answer Key Chapter 5 Lesson 5.5 Slope and Similar TrianglesĀ as a quick guide to resolve any of their doubts.

## Spectrum Math Grade 8 Chapter 5 Lesson 5.5 Slope and Similar Triangles Answers Key

The rate of change, or slope, of a line can be tested for constancy by using similar triangles.

To test if the slope of the line 6 is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to line 8 to create a set of right triangles.
Find the length of the legs for each set of triangles. 3 & 1 and 6 & 2
Test the leg lengths for proportionality. $$\frac{3}{1}$$ = $$\frac{6}{2}$$
3 Ć 2 = 6 and 6 Ć 1 = 6
These leg lengths are proportional, so the line has a constant slope.

Use similar right triangles to prove that each line has a constant slope.

Question 1.
a.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
2 & 1
Triangle 2 Legs:
4 & 2
Proportionality Test:
$$\frac{2}{1}$$ = $$\frac{4}{2}$$
2 Ć 2 = 4 and 4 Ć 1 = 4
These leg lengths are proportional, so the line has a constant slope.

b.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
1 & 2
Triangle 2 Legs:
2 & 4
Proportionality Test:
$$\frac{1}{2}$$ = $$\frac{2}{4}$$
2 Ć 2 = 4 and 4 Ć 1 = 4
These leg lengths are proportional, so the line has a constant slope.

Question 2.
a.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
3 & 2
Triangle 2 Legs:
6 & 4
Proportionality Test:
$$\frac{3}{2}$$ = $$\frac{6}{4}$$
3 Ć 4 = 12 and 6 Ć 2 = 12
These leg lengths are proportional, so the line has a constant slope.

b.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
3 & 2
Triangle 2 Legs:
6 & 4
Proportionality Test:
$$\frac{2}{3}$$ = $$\frac{4}{6}$$
3 Ć 4 = 12 and 6 Ć 2 = 12
These leg lengths are proportional, so the line has a constant slope.

Use similar right triangles to prove that each line has a constant slope.

Question 1.
a.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
4 & 1
Triangle 2 Legs:
8 & 2
Proportionality Test:
$$\frac{4}{1}$$ = $$\frac{8}{2}$$
4 Ć 2 = 8 and 8 Ć 1 = 8
These leg lengths are proportional, so the line has a constant slope.

b.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
3 & 2
Triangle 2 Legs:
6 & 4
Proportionality Test:
$$\frac{3}{2}$$ = $$\frac{6}{4}$$
3 Ć 4 = 12 and 6 Ć 2 = 12
These leg lengths are proportional, so the line has a constant slope.

Question 2.
a.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
3 & 1
Triangle 2 Legs:
6 & 2
Proportionality Test:
$$\frac{3}{1}$$ = $$\frac{6}{2}$$
3 Ć 2 = 6 and 6 Ć 1 = 6
These leg lengths are proportional, so the line has a constant slope.

b.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
1 & 1
Triangle 2 Legs:
2 & 2
Proportionality Test:
$$\frac{1}{1}$$ = $$\frac{2}{2}$$
1 Ć 2 = 2 and 1 Ć 2 = 2
These leg lengths are proportional, so the line has a constant slope.

Question 3.
a.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
2 & 1
Triangle 2 Legs:
4 & 2
Proportionality Test:
$$\frac{2}{1}$$ = $$\frac{4}{2}$$
2 Ć 2 = 4 and 4 Ć 1 = 4
These leg lengths are proportional, so the line has a constant slope.

b.

Triangle 1 Legs:
____ & _____
Triangle 2 Legs:
____ & _____
Proportionality Test:
____ = _____
Answer:

To test if the slope of the line is constant, draw a set of parallel lines that intersect the line.
Then, draw a line segment from each of the parallel lines to create a set of right triangles.
Find the length of the legs for each set of triangles.
Triangle 1 Legs:
3 & 1
Triangle 2 Legs:
6 & 2
Proportionality Test:
$$\frac{3}{1}$$ = $$\frac{6}{2}$$
3 Ć 2 = 6 and 6 Ć 1 = 6
These leg lengths are proportional, so the line has a constant slope.

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