Michael is 6 feet tall and is standing outside next to his younger sister. He notices that he can see both of their shadows and decides to measure each shadow. His shadow is 8 feet long and his sister's shadow is 5 feet long. How tall is Michael's sister?
Watch This
http://www.youtube.com/watch?v=LhEe0kB4QIs James Sousa: Indirect Measurement Using Similar Triangles
Guidance
If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional. There are three criteria for proving that triangles are similar:
 AA: If two triangles have two pairs of congruent angles, then the triangles are similar.
 SAS: If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.
 SSS: If three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.
Once you know that two triangles are similar, you can use the fact that their corresponding sides are proportional and their corresponding angles are congruent to solve problems. In the Examples and practice, you will consider many different applications of similar triangles.
Example A
a) Prove that the two triangles below are similar.
b) Use the Pythagorean Theorem to find
c) Use the fact that the triangles are similar to find the missing sides of
Solution: a) The triangles are similar by
b)
c)

ABDE=3→AB3=3→AB=9 
BCEF=3→BC33√=3→BC=93√
Example B
The triangles in Example A are called 306090 triangles because of their angles measures.
a) Explain why all 306090 triangles are similar.
b) Use the triangles from Example A to find the ratios between the three sides of any 306090 triangle.
c) Find the missing sides of the triangle below.
Solution: a) All 306090 triangles are similar by
b)
c) The side opposite the
Example C
Create similar triangles in order to solve for
Solution: Extend
Next, solve for
Concept Problem Revisited
The sun creates shadows at the same angle for both Michael and his sister. Assuming they are both standing up straight and making right angles with the ground, similar triangles are created.
Corresponding sides are proportional because the triangles are similar.
Cross multiply and solve for his sister's height. His sister is 3.75 feet tall.
Vocabulary
Two figures are similar if a similarity transformation will carry one figure to the other. Similar figures will always have corresponding angles congruent and corresponding sides proportional.
AA, or AngleAngle, is a criterion for triangle similarity. The AA criterion for triangle similarity states that if two triangles have two pairs of congruent angles, then the triangles are similar.
SAS, or SideAngleSide, is a criterion for triangle similarity. The SAS criterion for triangle similarity states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.
SSS, or SideSideSide, is a criterion for triangle similarity. The SSS criterion for triangle similarity states that if three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.
A 454590 triangle is an example of a special right triangle. The measures of its angles are 45, 45, and 90.
A 306090 triangle is an example of a special right triangle. The measures of its angles are 30, 60, and 90.
Guided Practice
1. Prove that all isosceles right triangles are similar.
2. Find the measures of the angles of an isosceles right triangle. Why are isosceles right triangles called 454590 triangles?
3. Use the Pythagorean Theorem to find the missing side of an isosceles right triangle whose legs are each length
4. Use what you have learned in #1#3 to find the missing sides of the right triangle below without using the Pythagorean Theorem.
Answers:
1. Consider two generic isosceles right triangles:
Two pairs of sides are proportional with a ratio of
2. The base angles of an isosceles triangle are congruent. If the vertex angle is
3. The missing side is the hypotenuse of the right triangle,
4. If one of the legs is 3, then the other leg is also 3, so
Practice
1. Explain why all 306090 triangles are similar.
2. The ratio between the sides of any 306090 triangle is ________:________:________.
Find the missing sides of each triangle:
3.
4.
5.
6. Explain why all 454590 triangles are similar.
7. The ratio between the sides of any 454590 triangle is ________:________:________.
Find the missing sides of each triangle:
8.
9.
10.
Use the figure below for #11 and #12.
11. Prove that
12. Find the value of
Use the figure below for #13#15.
13. Find
14. Find
15. Find