Rules of Pythagoras All Triangles: The sum of the three angles equals 180°. Right Triangles: One of its angles must be a right angle (90°). The side opposite of the right angle is called the hypotenuse (c). The other sides of the triangle are called the legs (a and b). Hypotenuse (c) leg 2 (b) leg 1 (a)

Pythagorean Theorem c a b If it is a right triangle: a2 + b2 = c2

Pythagorean Theorem and its Converse b If it is a right triangle then a2 + b2 = c2 The converse states that if a2 + b2 = c2 then it must be a right triangle

Right Triangle 17 8 Prove that the following triangle is a right triangle Be Careful! Assign c correctly! 15 If a2 + b2 = c2 then it is a right triangle 82 + 152 = 172 64 + 225 = 289 289 = 289 , it is a right triangle

Acute Triangle c a b A triangle with three acute (less than 90°) angles In non-right triangles a2 + b2 will not equal c2 Still use the Pythagorean Theorem to determine if the triangle is acute or obtuse If c2 < a2 + b2 then the triangle is acute

Acute Triangle 10 Prove that the following triangle is an acute triangle 7 8 If it was a right triangle then a2 + b2 = c2 72 + 82 ≠ 102 49 + 64 ≠ 100 113 > 100 or a2 + b2 > c2 Since 113 > 100 it is an acute triangle

Obtuse Triangle c a b A triangle with one obtuse (greater than 90°) angle In non-right triangles a2 + b2 will not equal c2 Still use the Pythagorean Theorem to determine if the triangle is acute or obtuse If c2 > a2 + b2 then the triangle is obtuse

Obtuse Triangle 10 Prove that the following triangle is an obtuse triangle 6 7 If it was a right triangle then a2 + b2 = c2 62 + 72 ≠ 102 36 + 49 ≠ 100 100 > 85 or c2 > a2 + b2 Since 100 > 85 it is an obtuse triangle

Practice Find x:

If c is the measure of the hypotenuse, find each missing side: Practice If c is the measure of the hypotenuse, find each missing side: 1. a = 12, b = 9, c = ? c = 15 2. a = 8, b = ?, c = 21 b = 19.4

Find the missing measure in each right triangle: Practice Find the missing measure in each right triangle: 1. 2. c = 12.6 x = 21

In the following right triangles determine the value of x: Practice In the following right triangles determine the value of x: 1. 2. 12 (x + 4) 10 x 2x (x - 2) x = 7.3 x = 5.4

Practice The sides of a triangle are listed below, determine whether the triangle is obtuse, acute or right. 1. 8, 9, 13 obtuse 2. 7, 12, 13 acute