Similar Triangles

The AAA Similarity Postulate

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

The AAA Similarity Postulate

If Then

The AA Similarity Theorem

If Then

Example 1

RI II NO, RI =8, RB=3x+4,ON=16, and OB=x+18

Find a. RB b. OB Ans. x=2 RB=10 OB=20

The SAS Similarity Theorem

If two sides of one triangle are proportional to the corresponding two sides of another triangle and their respective included angles are congruent, then the triangles are similar.

The SAS Similarity Theorem

If

Example 2

Are the two triangles similar? Justify your answer.

The SSS Similarity Theorem

If the sides of one triangle are proportional to the corresponding sides of a second triangle, then the triangles are similar.

Similar right triangles

The L-L Similarity Theorem If the legs of a right triangle are proportional to the corresponding legs of another right triangle, the right triangles are similar.

The L-L Similarity Theorem

If

Similar right triangles

The H-L Similarity Theorem If the hypotenuse and a leg of a right triangle are proportional to the corresponding hypotenuse and leg of another right triangle, then the right triangles are similar.

The H-L Similarity Theorem

If

Example 3

In the figure UA

Proportional Segments

The Proportional Segments Theorem

If a line intersects two sides of a triangle at distinct points and is parallel to the third side, the line divides the two sides in two proportional segments.

The Proportional Segments Theorem

𝑙𝑄𝑃

|| BC, then

Example 4

In AB||QR. If OA=5,

PA=2, and BR=10,

find PB.

Example 5

A flagpole 8m high casts a shadow of 12m, while a nearby building casts a shadow of 60m. How high is a building?

Proportional Segments

The Bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides.

Proportional Segments

If with AD an angle bisector, then

Example 6Find the value of x.