1. Special Right Triangles Keystone Geometry

2. Review: Parts of a Right Triangle

3. Special Types of Right Triangles
Special Types of Right Triangles A right triangle must have exactly one 90 degree angle. That leaves the two remaining angles to be acute and complementary. One type is Another type is

4. 45°-45°-90° Special Right Triangle
* A 45ｰ45ｰ90 triangle is an isosceles triangle with congruent legs. If the length of a leg is a, then the length of the hypotenuse is a times the square root of 2.

5. 45°-45°-90° Special Right Triangle
In a triangle 45°-45°-90° , the hypotenuse is times as long as a leg.
Example:
45°
45° cm
Hypotenuse
5 cm
Leg x
45°
5 cm
45° x
Leg

6. 30°-60°-90° Special Right Triangle
* In a 30ｰ60ｰ90 triangle, the shorter leg is opposite the 30 angle and the longer leg is opposite the 60 angle. The theorem says if the shorter leg has length a, then the hypotenuse has length 2a and the longer leg has length a times the square root of 3.

7. 30°-60°-90° Special Right Triangle
In a triangle 30°-60°-90° , the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.
Example:
Hypotenuse
30° 2X
Longer Leg
30°
10 cm X
5 cm
60°
60° X
5 cm
Shorter Leg

8. Example: Find the value of a and b.
b = 14 cm
60°
7 cm
30° 2x b
30 °
60°
a = cm a x
30°
Step 1: Find the missing angle measure.
30°-60°-90°
Step 2: Decide which special right triangle applies.
Step 3: Match the 30°-60°-90° pattern with the problem.
Step 4: From the pattern, we know that x = 7, b = 2x and a = x
Step 5: Solve for a and b

9. Example: Find the value of a and b.
b = cm
45°
7 cm
45° x b x
45 °
45°
a = 7cm a x
Step 1: Find the missing angle measure.
45°
Step 2: Decide which special right triangle applies.
45°-45°-90°
Step 3: Match the 45°-45°-90° pattern with the problem.
Step 4: From the pattern, we know that x = 7 , a = x, and b = x .
Step 5: Solve for a and b