This document discusses right triangles and similarity theorems. It defines a right triangle as having one 90 degree angle. The longest side is called the hypotenuse and the other two sides are called the legs. It states two right triangle similarity theorems: if two sides are proportional between triangles, or if one acute angle is congruent between triangles, then the triangles are similar. It also discusses special right triangles, noting that in 30-60-90 triangles specific side lengths are related and in 45-45-90 triangles the hypotenuse is √2 times the leg length. Examples are provided to demonstrate applying these concepts.

1. Right Triangle
Similarity

2. 01
Right Triangles and
Their Parts

3. Right Triangles are triangles that have a
right angle, or an angle measuring
exactly 90° . The longest side of a right
triangle is called a hypotenuse; the other
two sides are called the legs.
RIGHT
TRIANGLES

4. LEGS RIGHT ANGLE
HYPOTENUSE
RIGHT TRIANGLE
B
a
C
A B
c

5. Right Triangle
Similarity Theorems
02

6. Right Triangle Similarity Theorems
If any two sides of a right
triangle are proportional to
the corresponding sides of
another right triangle, then
the right triangles are
similar.
If an acute angle of a right
triangle is congruent to
another acute angle of another
right triangle, then the right
triangles are similar.

7. 03
Special Right
Triangle Theorems

8. Special Right Triangles Theorem
In 30°-60°-90° triangles, the side opposite the 90°
angle is the longest (hypotenuse), the leg opposite the
30° angle is one-half of the hypotenuse, and the leg
opposite the 60° angle is equal to the hypotenuse
multiplied by √3/2.

9. In 45°-45°-90° triangles, the legs
opposite the 45° angles are congruent
and each side is equal to the length of
the hypotenuse divided by √2.
This means that if a leg of a 45°-45°-
90° triangle is given, the hypotenuse is
just √2 times the length of a leg.
Special Right Triangles
Theorem

10. 04
Example
Problems

11. Are the two triangles
on the right similar?
If so, identify the
corresponding angles
and sides.
Example 1

13. Find the
lengths of all
the sides as
shown.
Example 2

15. Summary
A right triangle is a triangle
that has a right angle, which
is an angle that measures
90°. The longest side of a
right triangle is called the
hypotenuse; the other two
sides are called the legs.

16. Summary
Right Triangle Similarity Theorem
● If two sides of a right triangle are proportional to the
corresponding sides of another right triangle, then the
right triangles are similar.
● If an acute angle of a right triangle is congruent to an
acute angle of another right triangle, then the right
triangles are similar.

17. Summary
Special Right Triangles Theorem
● In 30°-60°-90° triangles, the side opposite the 90° angle is the
longest (hypotenuse), the leg opposite the 30° angle is half of the
hypotenuse, and the leg opposite the 60° angle is √3/2 times the
length of the hypotenuse or √3 times the length of the shorter leg.
● In a 45°-45°-90° triangle, the hypotenuse is √2 times the length
of a leg

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