Similar Triangles

Focus – Apply the properties of similar triangles and tomorrow, prove that triangles are similar.

Lesson 8-3

WA State Standards: G.3.A and G.3.B

Congruent Triangles

…have matching angles that are congruent. …have matching sides that are congruent.

A

B C

D

F

E

100°

35°

45°

115 ft

150 ft

100

ft

150 ft

115 ft

100

ft

35°

100°

45°

Congruent Triangles

…have matching angles that are congruent. …have matching sides that are congruent.

A

B C

D

F

E

100°

35°

45°

115 ft

150 ft

100

ft

150 ft

115 ft

100

ft

35°

100°

45°

m C m E

m B m F

m A m D

CB EF

AC DE

BA FD

Similar Triangles

IF and ONLY IF Vertices match up so corresponding angles

are congruent. Corresponding sides are in proportion.

30°30°

75° 75°

75° 75°

16

1212

16

6

8

Ratios of each side are4

3

Triangle Similarity Postulate

If two angles of one triangle are equal in measure to two angles of another triangle,

then the two triangles are similar.

AA (angle/angle) similarity

AA?

• You will also see:

• SAS

• ASA

• SSSKnowing these letters

will help with proofs later.

• Side/Angle/Side

• Angle/Side/Angle

• Side/Side/Side

Are they similar?

Only one angle is given as congruent. Two must be given to use Angle/Angle or AA Similarity.

Use Angle/Angle or AA Similarity. Two congruent angles show triangles are similar.

Are they similar?

Similar?

Find the missing side of each triangle to find two 30° angles and a 120° angle for each of these similar triangles.

120°

30°

30°

30°

Is ABC similar to AEF?

A

E F

CB

Sometimes it helpsto separate the two

triangles and look at eachangle separately.

Find the missing side

We previously determined that these triangles are similar. We can set up ratios to find the missing side.

Start with a label on top and bottom.

120°

30°

30°

30°

7 ft

21 ft

28 ft

n ft

short

long 21

28

7

n

In today’s lesson…

• We found that congruent triangles have both congruent angles and sides.

• Similar triangles have congruent angles.

• We can use the AA similarity to determine if triangles are similar.

• We can use ratios to determine a missing side’s length when similar triangles are used.

WA State Standards: G.3.A and G.3.B

Assign: 453: 4-8; 12-13 457: 1-4

This statue can be seen in downtown Seattle in the Pacific Place Mall on the main level.

Day Two

Yesterday, we found that….• We found that congruent triangles have both

congruent angles and sides.• Similar triangles have congruent angles.• We can use the AA similarity to determine if

triangles are similar.

Today’s Focus-Prove that triangles are similar.

Overlapping Triangles

It is sometimes useful to redraw as separate triangles to name the congruent sides and angles of those triangles.

BC

A

CB

AF

AE

EF

A

FE

AC

AB

Is ABC similar to AEF?If so, what Is the missing side?

y

15

16

x 12

12

A

B

C

E

F

It often helps to separate the two attached triangles.

y

15

16

x

12

12

Prove: A line drawn from a point on one side of a triangle parallel to another side forms a triangle similar to the original triangle.

Did you notice that the words corresponding occur with parallel lines and triangles?

Corresponding Angles in triangles are different than when working with parallel lines, but in both cases are congruent.

Given: ;Prove:

1.

2.

3.

1. Given

2. If two || lines are intersected by a transversal, then corresponding angles are = in measure

3. AA Similarity

||DE BC

ABC ||DE BCADE ABC

ADE ABC

m AED m ACB m ADE m ABC

WA State Standards: G.3.A and G.3.B

B

A

D

C

E

9. Given: Prove:

1.

2.

3.

1.Given

2.Alternate Interior Angles from Transversal/Parallel Lines Theorem

3.AA Similarity Theorem

||AB DEABC EDC

A

E

D

B

C

||AB DEm A m E

ABC EDC

m B m D

WA State Standards: G.3.A and G.3.B

Overlapping Similar Triangles Theorem

If a line is drawn from a point on one side of a triangle parallel to another side,

…then it forms a triangle similar to the original triangle.

Solve by using proportions

4 ft

6 ft

5 ft

x

4 6

5 x

30 17

4 2

Assign: 453: 9; 14, 16, 21a, 23, 26 457: 5-7