This document provides information and examples for determining congruence and similarity using slopes of lines and triangles. It explains that the ratio of the rise to the run of two slope triangles formed by a line is equal to the slope of the line. It also gives examples of finding the slope of a roof, stairs, and verifying that the slope is the same between points on the line. Finally, it discusses that right triangles that have their hypotenuses on the same line in a coordinate plane are called slope triangles, and if two triangles are slope triangles then they are similar.

1. Course 3, Lesson 7-6
Write each number in standard form.
1. At the same time a 4-foot fencepost casts a 6-foot shadow, a
cellular tower casts a 50-foot shadow. How tall is the cellular
tower to the nearest tenth foot?
2. If the man in the picture
is 6 feet tall, how tall is
the tree?
3. A mother and daughter are
standing next to each other.
The mother is 160 centimeters tall and has a shadow that is 100
centimeters long. The daughter’s shadow is 50 centimeters long.
How tall is the daughter?

2. Course 3, Lesson 7-6
ANSWERS
1. 33.3 feet
2. 24 feet
3. 80 centimeters

3. HOW can you determine
congruence and similarity?
Geometry
Course 3, Lesson 7-6

4. Course 3, Lesson 7-6 Common Core State Standards © Copyright 2010. National Governors Association Center for
Best Practices and Council of Chief State School Officers. All rights reserved.
• 8.EE.6
Use similar triangles to explain why the slope m is the same between
any two distinct points on a non-vertical line in the coordinate plane;
derive the equation y = mx for a line through the origin and the equation
y = mx + b for a line intercepting the vertical axis at b.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
Geometry

5. To
• verify that the slope m of a line is the same
between any two points on a coordinate
plane using slope triangles
Course 3, Lesson 7-6
Geometry

6. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
1. Write a proportion comparing the rise to the
run for each of the similar slope triangles
shown. Then find the numeric value.
AC • DE = BE • BC
Corresponding sides of similar
triangles are proportional.
Find the cross products.
Division Property of Equality
Simplify.
AC = 6, BC = 3, BE = 4, DE = 2

7. Answer
Need Another Example?
Graph ABC with vertices A(−4, 2), B(−4, −2), and
C(−2, −2), and CDF with vertices C(−2, −2),
D(−2, −4), and F(−1, −4). Then write a proportion
comparing the rise to the run for each of the similar
slope triangles and find the numeric value.
;

8. Course 3, Lesson 7-6
Geometry
Words The ratio of the rise to the run of two slope triangles formed by
a line is equal to the slope of the line.
Example

9. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
2. The pitch of a roof refers to the slope of
the roof line. Choose two points on the
roof and find the pitch of the roof shown.
Then verify that the pitch is the same by
choosing a different set of points.
The pitch of the roof is . Verify that the pitch is the same using
two other points.
Formula for slope
Use the points S and R.
(x1, y1) = (8, 6) and (x2, y2) = (12, 8)
Formula for slope
Simplify.
Use the points U and T.
(x1, y1) = (2, 3) and (x2, y2) = (0, 2)
7 Simplify. The pitch is the same.

10. Answer
Need Another Example?
Choose two points along the stairs and find the
slope of the stairs. Then verify that the slope is
the same by choosing a different set of points.
m = 1; The other slope should equal 1.

11. How did what you learned
today help you answer the
HOW can you determine
congruence and similarity?
Course 3, Lesson 7-6
Geometry

12. How did what you learned
today help you answer the
HOW can you determine
congruence and similarity?
Course 3, Lesson 7-6
Geometry
Sample answers:
• In a coordinate plane, right triangles that have their
hypotenuses on the same line are called slope
triangles.
• If two triangles are slope triangles, then they are
similar.

13. Graph and connect the following
coordinates: (0, 1), (5, 4), and (10, 7).
Find the slope. Then use the slope
and the coordinates given to draw a
congruent right triangle on the same grid.
Finally, create a triangle that is similar
to the two triangles you just drew.
Course 3, Lesson 7-6
Ratios and Proportional RelationshipsFunctionsGeometry