1. Special Right Triangles

2. What are Special Right Triangles? There are 2 types of Right triangles that are considered special. We will talk about only one of these today.

3. Imagine a square. What makes a square…square? All 4 sides congruent All 4 angles congruent What do the 4 angles add up to? 360 0 So each angle is? 90 0

4. Now divide this square with a diagonal. What kind of triangles are formed?? 2 congruent sides and a right angle in each triangle. ISOSCELES RIGHT TRIANGLES!!!

5. Each triangle has a right angle and 2 congruent sides. If a triangle has 2 congruent sides, then it must have 2 congruent angles. What are the 2 angles? This is the first special right triangle. A 45-45-90 triangle. 45 0

6. Standard: MM2G1b Determine the lengths of the sides of 45-45-90 Triangles Essential Question: What patterns can I use to find the lengths of the sides of a right triangle?

7. Parts of a 45-45-90 Right Triangle Hypotenuse Leg The legs of a 45-45-90 are congruent!!!!

8. Performance Task Complete Performance Task

9. 45º - 45º - 90º Theorems IN A 45-45-90 Δ THE HYPOTENUSE IS TIMES AS LONG AS EACH LEG 45 o x x __

10. Example 10 C A B Find BC and AB 10 BC and AC are equal, so BC = 10. AB is the Hypotenuse and is times AC. AB is

11. Ex: find x 5 5 x 45 __ x=5 45 12 x

12. EXAMPLE 1 Find hypotenuse length in a 45-45-90 triangle o o o Find the length of the hypotenuse. a. SOLUTION hypotenuse = leg 2 = 8= 8 2 Substitute. 45-45-90 Triangle Theorem o o o By the Triangle Sum Theorem, the measure of the third angle must be 45 º. Then the triangle is a 45 º -45 º - 90 º triangle, so by Theorem 7.8, the hypotenuse is 2 times as long as each leg. a.

13. EXAMPLE 2 Find hypotenuse length in a 45-45-90 triangle o o o hypotenuse = leg 2 Substitute. 45-45-90 Triangle Theorem o o o = 3 22 = 3 2 Product of square roots = 6 Simplify. b. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. 45 - 45 - 90 o o o Find the length of the hypotenuse. b.

14. EXAMPLE 3 Find leg lengths in a 45-45-90 triangle o o o Find the lengths of the legs in the triangle. SOLUTION By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. 45 - 45 - 90 o o o hypotenuse = leg 2 Substitute. 45-45-90 Triangle Theorem o o o 2 5 = x= x 2 2 5 2 = 2 x 2 5 = x Divide each side by 2 Simplify.

15. EXAMPLE 4 Standardized Test Practice SOLUTION By the Corollary to the Triangle Sum Theorem, the triangle is a triangle. 45 - 45 - 90 o o o

16. EXAMPLE 4 Standardized Test Practice hypotenuse = leg 2 Substitute. 45-45-90 Triangle Theorem o o o = 252 WX The correct answer is B.

17. GUIDED PRACTICE for Examples 1, 2, and 3 Find the value of the variable. 1.2.3. ANSWER 2 2 8 2

18. GUIDED PRACTICE for Examples 1, 2, and 3 4. Find the leg length of a 45°- 45°- 90° triangle with a hypotenuse length of 6. 3 2 ANSWER