Special Right Triangles b A C B a c 45 Consider: Right ΔABC with <A = <B = 45° Given Inverse ITT Pythagorean Theorem Substitution (2, 3) Addition Simplifying Radicals <A = <B a = b c2 = a2 + b2 c2 = a2 + a2 c2 = 2a2 c =

45° - 45° - 90° Triangles 45-45-90 Theorem: In a 45-45-90 triangle, the hypotenuse is √2 times as long as a leg Ex 1) Find x Ex 2) Find y Ex 3) Find z m z 12 x y 6

Application Questions Find the length of a diagonal of a square with sides 10 in long. Find the length of a side of square whose diagonal is 4 cm. Solve for y: Solution: 10√2 Solution: 2√2 45° ☐ 6 10 y Solution: 8√2

Consider: Right ΔABC with <A = 30° and <B = 60° ☐ B C c a D c a Consider: Right ΔABC with <A = 30° and <B = 60° 30 60 <A = 30°, <B = 60 Draw CD s.t. CD = a, AD = c ΔACD ≅ ΔACB <D = <B = <DAB = 60° ΔDAB is equilateral c = 2a c2 = a2 + b2 (2a)2 = a2 + b2 4a2 = a2 + b2 b2 = 3a2 b = a√3 Given Ruler Postulate SAS CPCTC If equiangular then equilateral Def of equilateral Pythagorean Theorem Substitution (6,7) Multiplication Subtraction Square Root Property

30° - 60° - 90° Triangles x 2x x√3 ☐ 30 30-60-90 Theorem: In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. Ex 1) Find x and y Ex 2) Find a and b Ex 3) Find m and n triangle is eqilateral 5 y x 30 b 8 30 60 a m 16 n m = 8√3 n = 16 x = 10 y = 5√3 a = 4√3 b = 4