1. 8.4A Similar Triangles and Slope

2. Objectives/Assignment
Identify similar triangles.
Use similar triangles to help in developing an understanding of slope, m, given as a rate comparing the change in y-values and x-values.

3. Identifying Similar Triangles
In this lesson, you will continue the study of similar polygons by looking at the properties of similar triangles.
In this lesson, you will use similar triangles and polygons to develop an understanding of slope

4. Ex. 1: Writing Proportionality Statements
In the diagram, ∆BTW ~ ∆ETC.
Write the statement of proportionality.
Find mTEC.
Find ET and BE.
34°
79°
*****Pay close attention to where each segment is located.

5. Ex. 1: Writing Proportionality Statements
In the diagram, ∆BTW ~ ∆ETC.
Write the statement of proportionality.
34° ET TC CE = = BT TW WB
79°
*****Pay close attention to how this ratio is set up. Look at where the segments are located.

6. Ex. 1: Writing Proportionality Statements
In the diagram, ∆BTW ~ ∆ETC.
Find mTEC.
B  TEC, SO mTEC = 79°
34°
79°

7. Ex. 1: Writing Proportionality Statements
In the diagram, ∆BTW ~ ∆ETC.
Find ET and BE.
34° CE ET
Write proportion. = WB BT 3 ET
Substitute values. = 12 20
3(20) ET
Multiply each side by 20. =
79° 12 5 = ET
Simplify.
Because BE = BT – ET, BE = 20 – 5 = 15. So, ET is 5 units and BE is 15 units.

8. Angle-Angle Similarity Postulate
If two angles of one triangle are congruent to the two angles of another triangle, then the two triangles are similar.
If JKL  XYZ and KJL  YXZ, then ∆JKL ~ ∆XYZ.

9. Ex. 3: Why a Line Has Only One Slope
∆ ADF ~ ∆ BCE
Use the properties of similar triangles to explain why any two points on a line can be used to calculate slope. Find the slope of the line using both pairs of points shown.

10. Ex. 3: Why a Line Has Only One Slope
By the AA Similarity Postulate, ∆BEC ~ ∆AFD, so the ratios of corresponding sides are the same. In particular, CE BE
By a property of proportions, = DF AF CE DF = BE AF

11. Ex. 3: Why a Line Has Only One Slope
The slope of a line is the ratio of the change in y to the corresponding change in x. The ratios
Represent the slopes of BC and AD, respectively.
and CE BE DF AF

12. Ex. 3: Why a Line Has Only One Slope
Because the two slopes are equal, any two points on a line can be used to calculate its slope. You can verify this with specific values from the diagram.
3-0 3 =
Slope of BC
4-2 2
6-(-3) 9 3 = =
Slope of AD
6-0 6 2