11.5 Similar Triangles11.5 Similar TrianglesIdentifying Corresponding Identifying Corresponding Sides of Similar TrianglesSides of Similar Triangles

By: Shaunta Gibson

Similar Triangles Similar Triangles are triangles that have are triangles that have the same shape but not necessarily the same size.the same shape but not necessarily the same size.

A

B

CD

E

F

In the diagram above, triangle ABC is equal to triangle DEF. We write it as ABC ~ DEF. In the diagram, each angle of ABC corresponds to an angle of DEF as follows:

Similar Triangles Similar Triangles are triangles that have are triangles that have the same shape but not necessarily the same size.the same shape but not necessarily the same size.

A

B

C D

E

F

angle A = angle D angle B = angle E angle C = angle F

Also, each side of ABC corresponds to a side of DEF

AB corresponds to DE BC corresponds to EF AC corresponds to DF

In similar triangles, In similar triangles, corresponding sides corresponding sides are the sides opposite the equal angles.are the sides opposite the equal angles.

* * When we write that two angles are similar, we name When we write that two angles are similar, we name them so that the order of corresponding angles in them so that the order of corresponding angles in both triangles is the same. both triangles is the same.

triangle ABC ~ triangle DEF

A

B

C D

E

F

Triangle RST ~ triangle XYZ. Name Triangle RST ~ triangle XYZ. Name the corresponding sides of these the corresponding sides of these triangles.triangles.

Because RST ~ XYZ that means angle R = angle X, angle Because RST ~ XYZ that means angle R = angle X, angle S = angle Y, and angle T = angle Z.S = angle Y, and angle T = angle Z.

Now we write the following:Now we write the following:

Angle R = Angle X, so ST corresponds to YZ.Angle R = Angle X, so ST corresponds to YZ.

Angle S = Angle Y, so RT corresponds to XZ.Angle S = Angle Y, so RT corresponds to XZ.

Angle T = Angle Z, so RS corresponds to XY.Angle T = Angle Z, so RS corresponds to XY.

RS

T

X Y

Z

In similar triangles, corresponding sides are in In similar triangles, corresponding sides are in proportion: that is, the ratios of their length are proportion: that is, the ratios of their length are equal. As shown below in the example triangle equal. As shown below in the example triangle ABC ~ DEF, therefore we have the followingABC ~ DEF, therefore we have the following

AB BC ACAB BC AC

DE EF DFDE EF DF

6 8 4 186 8 4 18

3 4 2 93 4 2 9 A

B

C

8 cm

4 cm

6 cm

D

E

F

3 cm

2 cm

4 cm 2

1

Finding the Missing Sides of Similar Finding the Missing Sides of Similar TrianglesTriangles

• To find a missing side of similar trianglesTo find a missing side of similar triangles

1.) 1.) write the ratios of the lengths of the write the ratios of the lengths of the correspondingcorresponding

sidessides

2.)2.) write a proportion using a ratio with known write a proportion using a ratio with known terms terms

and a ratio with an unknown termand a ratio with an unknown term

3.)3.) solve the proportion for the unknown term solve the proportion for the unknown term

In the following diagram, triangle In the following diagram, triangle TAP ~ triangle RUN. Find x.TAP ~ triangle RUN. Find x.

Because TAP ~ RUN, we write the ratios of the lengths of Because TAP ~ RUN, we write the ratios of the lengths of the corresponding sides.the corresponding sides.

T

A

P R U

N15 cm

30 cm

x

18 cm

12 cm

9 cm

15 x

9 12

x 30

12 18 or

Now cross multiply, then divide to Now cross multiply, then divide to get the length of AP.get the length of AP.

T

A

P R U

N15 cm

30 cm

x

18 cm

12 cm

9 cm

15 x

9 12

9x = 180: x = 20 so x, or the length of AP, is 20 cm

THANK YOUTHANK YOU

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