FeatureLesson Geometry Lesson Main (For help, go to Lessons 4-2 and 4-3.) Name the postulate or...

FeatureLesson

GeometryGeometry

LessonMain

(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.

Lesson 7-3

Proving Triangles SimilarProving Triangles Similar

Check Skills You’ll Need


7-3

FeatureLesson

GeometryGeometry

LessonMain

Lesson 7-3


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


7-3

FeatureLesson

GeometryGeometry

LessonMain

FeatureLesson

GeometryGeometry

LessonMain

Lesson 7-3


Notes

7-3

FeatureLesson

GeometryGeometry

LessonMain

Lesson 7-3


Notes

7-3

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.

FeatureLesson

GeometryGeometry

LessonMain

Use the trapezoids below for Exercises 1–3. DFHN ~ BMLP. Complete each statement.

1. mH = ?

2. x = ?

3. mD = ?

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.

74

49

99

16 in. by 24 in.

yes; 1 : 2

≈ 12.4 ft

Lesson 7-2

Similar PolygonsSimilar Polygons

Lesson Quiz

7-3

FeatureLesson

GeometryGeometry

LessonMain

FeatureLesson

GeometryGeometry

LessonMain

MX AB. Explain why the triangles are similar.

Write a similarity statement.

AMX ~ BKX by the Angle-Angle Similarity Postulate (AA ~ Postulate).

Lesson 7-3


Because MX AB, AXM and BXK are both right angles, so AXM BXK.

A B because their measures are equal.

Quick Check

Additional Examples

7-3

Using the AA ~ Postulate

FeatureLesson

GeometryGeometry

LessonMain

Explain why the triangles must be similar.

Write a similarity statement.

Therefore, YVZ ~ WVX by the Side-Angle-Side Similarity Theorem(SAS Similarity Theorem).

Lesson 7-3


= = and = = , so corresponding sides are proportional. VYVW

1224

12

VZVX

1836

12

YVZ WVX because they are vertical angles.

Quick Check

Additional Examples

7-3

Using Similarity Theorems

FeatureLesson

GeometryGeometry

LessonMain

ABCD is a parallelogram. Find WY.

Use the properties of similar triangles to find WY.

WY = 5 Solve for WY.104

WY = 12.5

Lesson 7-3


WZWX= Corresponding sides of ~ triangles are proportional.WY

WA

10 4= Substitute.WY

5

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.

Quick Check

Additional Examples

7-3

Finding Lengths in Similar Triangles

FeatureLesson

GeometryGeometry

LessonMain

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.

Use similar triangles to find the height of the tree.

Lesson 7-3


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 mJOM = mTRM, and therefore JOM TRM.

The light reflects off a mirror at the same angle at which it hits the mirror, so JMO TMR.

Additional Examples

7-3

Real-World Connection

FeatureLesson

GeometryGeometry

LessonMain

(continued)

JOM ~ TRM AA ~ Postulate

The tree is 40 ft tall.

Lesson 7-3


RMOM

= Corresponding sides of ~ triangles are proportional. TRJO

24 3

= Substitute. TR 5

TR = 40

= 5 Solve for TR. 24 3

TR 5

Quick Check

Additional Examples

7-3

FeatureLesson

GeometryGeometry

LessonMain

Are the triangles similar? If so, write a similarity statement and name the postulate or theorem you used. If not, explain.

1. 2.

3. 4. 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

MRT ~ XRW by AA ~ Postulate

16

31 ft

Lesson 7-3


Lesson Quiz

7-3

FeatureLesson Geometry Lesson Main (For help, go to Lessons 4-2 and 4-3.) Name the postulate or...

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