Students can use the **Spectrum Math Grade 8 Answer Key** **Chapter 5 Lesson 5.5 Slope and Similar Triangles**Ā 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.