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**Special Right Triangles Keystone Geometry**

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**Review: Parts of a Right Triangle**

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**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 45º-45º-90º Another type is 30º-60º-90º

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**45º- 45º- 90º Special Right Triangle**

* A 45ºｰ45ºｰ90º 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.

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**45º- 45º- 90º Special Right Triangle**

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

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**30º- 60º- 90º Special Right Triangle**

* In a 30ºｰ60ºｰ90º triangle, the shorter leg is opposite the 30º angle and the longer leg is opposite the 60º 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.

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**30º- 60º- 90º Special Right Triangle**

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

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**Example: Find the value of a and b.**

b = 14 cm 60º 7 cm 30º 2x b 30º° 60º a = cm a x Step 1: Find the missing angle measure. 30° Step 2: Decide which special right triangle applies. 30º- 60º- 90º Step 3: Match the 30º- 60º- 90º 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

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**Example: Find the value of a and b.**

b = cm 45º 7 cm 45º b x 45 ° 45º a = 7cm a x Step 1: Find the missing angle measure. 45° Step 2: Decide which special right triangle applies. 45º- 45º- 90º Step 3: Match the 45º - 45º- 90º 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

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