1. SIMILAR TRIANGLES

2. Similar triangles
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

3. Warm-Up
1. In ABC and XZW, m A = m X and m B = m Z. What can you conclude about m C and m W?
ANSWER
They are the same.
2. Solve = x 18 54 9
ANSWER
108

4. Warm-Up
ABC DEF. Find x. ~
ANSWER 10

5. Example 1
Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning.

6. Example 1
SOLUTION
Because they are both right angles, D and G are congruent.
By the Triangle Sum Theorem, 26° + 90° + m E = 180°, so m E = 64°. Therefore, E and H are congruent.
So, ∆CDE ~ ∆KGH by the AA Similarity Postulate.

7. Example 2
Show that the two triangles are similar. a.
∆ABE and ∆ACD
SOLUTION a.
You may find it helpful to redraw the triangles separately.
Because m ABE and m C both equal 52°, ABE C. By the Reflexive Property, A A.
So, ∆ ABE ~ ∆ ACD by the AA Similarity Postulate.

8. Example 2
Show that the two triangles are similar. b.
∆SVR and ∆UVT
SOLUTION
You know SVR  UVT by the Vertical Angles Congruence Theorem. The diagram shows RS ||UT so S  U by the Alternate Interior Angles Theorem. b.
So, ∆SVR ~ ∆UVT by the AA Similarity Postulate.

9. Guided Practice
Show that the triangles are similar. Write a similarity statement. 1.
∆FGH and ∆RQS
In each triangle all three angles measure 60°, so by the AA similarity postulate, the triangles are similar ∆FGH ~ ∆QRS.
ANSWER

10. Guided Practice
Show that the triangles are similar. Write a similarity statement. 2.
∆CDF and ∆DEF
Since m CDF = 58° by the Triangle Sum Theorem and m DFE = 90° by the Linear Pair Postulate the two triangles are similar by the AA Similarity Postulate; ∆CDF ~ ∆DEF.
ANSWER

11. Guided Practice 3.
REASONING Suppose in Example 2, part (b), SR TU. Could the triangles still be similar? Explain.
Yes; if S T, the triangles are similar by the AA Similarity Postulate.
ANSWER

12. Example 3

13. Example 3
SOLUTION
The flagpole and the woman form sides of two right triangles with the ground, as shown below. The sun’s rays hit the flagpole and the woman at the same angle. You have two pairs of congruent angles, so the triangles are similar by the AA Similarity Postulate.

14. The flagpole is 80 feet tall. The correct answer is C. ANSWER
Example 3
You can use a proportion to find the height x. Write 5 feet 4 inches as 64 inches so that you can form two ratios of feet to inches.
x ft
64 in.
50 ft
40 in. =
Write proportion of side lengths.
40x = 64(50)
Cross Products Property
x = 80
Solve for x.
The flagpole is 80 feet tall.
The correct answer is C.
ANSWER

15. Guided Practice 4.
WHAT IF? A child who is 58 inches tall is standing next to the woman in Example 3. How long is the child’s shadow?
ANSWER
36.25 in.

16. Guided Practice 5.
You are standing in your backyard, and you measure the lengths of the shadows cast by both you and a tree. Write a proportion showing how you could find the height of the tree.
tree height
your height =
length of your shadow
length of shadow
SAMPLE ANSWER

17. Lesson Quiz
Determine if the two triangles are similar. If they are write a similarity statement. 1.
ANSWER
Yes; ABE ~ ACD

18. Lesson Quiz
Determine if the two triangles are similar. If they are write a similarity statement. 2.
ANSWER no

19. Lesson Quiz
3. Find the length of BC.
ANSWER
7.5

20. Lesson Quiz
4. A tree casts a shadow that is 30 feet long. At the same time a person standing nearby, who is five feet two inches tall, casts a shadow that is 50 inches long. How tall is the tree to the nearest foot?
37 ft
ANSWER