1. Bell Ringer

2. A right triangle with angles measure of 45,45, and 90 are called 45-45-90 triangles

3. By the 45° –45° –90° Triangle Theorem, the length of the
Example 1
Find Hypotenuse Length
Find the length x of the hypotenuse in the 45° –45° –90° triangle shown at the right.
SOLUTION
By the 45° –45° –90° Triangle Theorem, the length of the
hypotenuse is the length of a leg times . 2
45° –45° –90° Triangle Theorem
hypotenuse = leg 2
Substitute.
= 3 2
ANSWER
The length of the hypotenuse is 2 3

4. Find the length x of each leg in the
Example 2
Find the length x of each leg in the
45° –45° –90° triangle shown at the right.
Find Leg Length
SOLUTION
By the 45° –45° –90° Triangle Theorem, the length of the
hypotenuse is the length of a leg times . 2
45° –45° –90° Triangle Theorem
hypotenuse = leg 2
Substitute.
= x 7 2 = x 2 7
Divide each side by .
7 = x
Simplify.
ANSWER
The length of each leg is 7. 4

5. Now You Try  Find the value of x. 1. ANSWER 4 2 2. ANSWER 5 2 3.
Find Hypotenuse and Leg Lengths
Find the value of x. 1.
ANSWER 4 2 2.
ANSWER 5 2 3.
ANSWER 3
ANSWER 6 4.

6. By the Triangle Sum Theorem, x° + x° + 90° = 180°.
Example 3
Identify 45° –45° –90° Triangles
Determine whether there is enough information to conclude that the triangle is a 45° –45° –90° triangle. Explain your reasoning.
SOLUTION
By the Triangle Sum Theorem, x° + x° + 90° = 180°.
So, 2x° = 90°, and x = 45.
ANSWER
Since the measure of each acute angle is 45°, the triangle is a 45° –45° –90° triangle. 6

7. You can use the 45° –45° –90° Triangle Theorem to find the value of x.
Example 4
Show that the triangle is a 45° –45° –90° triangle. Then find the value of x.
Find Leg Length
SOLUTION
The triangle is an isosceles right triangle. By the Base Angles Theorem, its acute angles are congruent. From the result of Example 3, this triangle must be a 45° –45° –90° triangle.
You can use the 45° –45° –90° Triangle Theorem to find the value of x.
45° –45° –90° Triangle Theorem
hypotenuse = leg 2
Substitute.
= x 5 2 7

8. Use a calculator to approximate.
Example 4
Find Leg Length
Divide each side by . 2 = x 5
Simplify.
= x 5 2
3.5 ≈ x
Use a calculator to approximate. 8

9. Now You Try 
Find Leg Lengths
Show that the triangle is a 45° –45° –90° triangle. Then find the value of x. Round your answer to the nearest tenth. 5.
ANSWER
x = ≈ 5.7.
The triangle is an isosceles right triangle. By the Base Angles Theorem, its acute angles are congruent. From the result of Example 3, the triangle is a
45° –45° –90° triangle. 8 2

10. Checkpoint Now You Try 
Find Leg Lengths
Show that the triangle is a 45° –45° –90° triangle. Then find the value of x. Round your answer to the nearest tenth. 6.
ANSWER
The triangle has congruent acute angles. By Example 3, the triangle is a 45° –45° –90°
triangle.
x = ≈ 8.5. 12 2