1. Proving Triangles Similar
Lesson 7-3
Check Skills You’ll Need
(For help, go to Lessons 4-2 and 4-3.)
Name the postulate or theorem you can use to prove the triangles congruent.
1. 2. 3.
Check Skills You’ll Need
7-3

2. Proving Triangles Similar
Lesson 7-3
Check Skills You’ll Need
1. All corresponding sides are marked congruent, so Side-Side-Side, or SSS postulate
2. Two sides and an included angle of one are congruent to two sides and an included angle of the other, so Side-Angle-Side, or SAS Postulate
3. Two angles and an included side of one are congruent to two angles and an included side of the other, so Angle-Side-Angle, or ASA Postulate
Solutions
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5. Proving Triangles Similar
Lesson 7-3
Notes
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6. Proving Triangles Similar
Lesson 7-3
Notes
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7. Proving Triangles Similar
Lesson 7-3
Notes
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8. Proving Triangles Similar
Lesson 7-3
Notes
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9. Proving Triangles Similar
Lesson 7-3
Notes
Indirect measurement uses similar triangles and measurements to find distances that are difficult to measure directly.
One method of indirect measurement uses the fact that light reflects off a mirror at the same angle at which it hits the mirror. A second method uses the similar triangles that are formed by certain figures and their shadows.
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10. Similar Polygons
Lesson 7-2
Lesson Quiz
Use the trapezoids below for Exercises 1–3. DFHN ~ BMLP. Complete each statement.
1. mH = ?
2. x = ?
3. mD = ? 74 49 99
4. A 4-in. by 6-in. drawing is enlarged to fit on a poster that measures 20 in. by 24 in. What are the dimensions of the largest drawing possible?
5. A rectangle with a perimeter 20 cm has a side 4 cm long. A rectangle with perimeter 40 cm has a side 8 cm long. Determine whether the rectangles are similar. If they are, give the similarity ratio. If they are not, explain.
6. The longer side of the golden rectangle is 20 ft. Find the length of the shorter side, rounded to the nearest tenth.
16 in. by 24 in.
yes; 1 : 2
≈ 12.4 ft
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14. Proving Triangles Similar
Lesson 7-3
Additional Examples
Using the AA ~ Postulate
MX AB. Explain why the triangles are similar. Write a similarity statement.
Because MX AB, AXM and BXK are both right angles, so
AXM BXK.
A B because their measures are equal.
AMX ~ BKX by the Angle-Angle Similarity Postulate (AA ~ Postulate).
Quick Check
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15. Proving Triangles Similar
Lesson 7-3
Additional Examples
Using Similarity Theorems
Explain why the triangles must be similar. Write a similarity statement.
YVZ WVX because they are vertical angles.
= = and = = , so corresponding sides are proportional. VY VW 12 24 1 2 VZ VX 18 36
Therefore, YVZ ~ WVX by the Side-Angle-Side Similarity Theorem (SAS Similarity Theorem).
Quick Check
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16. Proving Triangles Similar
Lesson 7-3
Additional Examples
Finding Lengths in Similar Triangles
ABCD is a parallelogram. Find WY.
Because ABCD is a parallelogram, AB || DC.
XAW ZYW and AXW YZW because parallel lines cut by a transversal form congruent alternate interior angles.
Therefore, AWX ~ YWZ by the AA ~ Postulate.
Use the properties of similar triangles to find WY. WZ WX =
Corresponding sides of ~ triangles are proportional. WY WA 10 4 =
Substitute. WY 5
WY =  5
Solve for WY. 10 4
WY = 12.5
Quick Check
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17. Proving Triangles Similar
Lesson 7-3
Additional Examples
Real-World Connection
Joan places a mirror 24 ft from the base of a tree. When she stands 3 ft from the mirror, she can see the top of the tree reflected in it. If her eyes are 5 ft above the ground, how tall is the tree?
Draw the situation described by the example.
TR represents the height of the tree, point M represents the mirror, and point J represents Joan’s eyes.
Both Joan and the tree are perpendicular to the ground, so mJOM = mTRM, and therefore JOM TRM.
The light reflects off a mirror at the same angle at which it hits the mirror, so JMO TMR.
Use similar triangles to find the height of the tree.
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18. Proving Triangles Similar
Lesson 7-3
Additional Examples
(continued)
JOM ~ TRM
AA ~ Postulate RM OM =
Corresponding sides of ~ triangles are proportional. TR JO 24 3 =
Substitute. TR 5
TR = 40
=  5
Solve for TR. 24 3 TR 5
The tree is 40 ft tall.
Quick Check
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19. Proving Triangles Similar
Lesson 7-3
Lesson Quiz
Are the triangles similar? If so, write a similarity statement and name the
postulate or theorem you used. If not, explain.
1. 2.
Find the value of x.
5. When a 6 ft tall man casts a shadow 18 ft long, a nearby tree casts a shadow 93 ft long. How tall is the tree?
The congruent angle is not included between the proportional sides, so you cannot conclude that the triangles are similar.
ABX ~ CDX by SAS ~ Theorem 16
MRT ~ XRW by AA ~ Postulate
31 ft
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