Discover the properties of Isosceles Triangles. Lesson 4.2 Discover the properties of Isosceles Triangles. HOMEWORK: Lesson 4.2/1-11 QUIZ Tomorrow: 4.1 – 4.2

Isosceles Triangle at least two sides have the same length 9 in 9 in 3 miles 3 miles 3 miles 4 in

Properties of an Isosceles Triangle 1 Has at least 2 equal sides Has 2 equal angles Has 1 line of symmetry

Equilateral Triangle Isosceles Triangle with all three sides are congruent 7 ft 7 ft 7 ft

Parts of an Isosceles Triangle: The vertex angle is the angle between two congruent sides

Parts of an Isosceles Triangle: The base angles are the angles opposite the congruent sides

Parts of an Isosceles Triangle: The base is the side opposite the vertex angle

Isosceles Triangle Conjecture If a triangle is isosceles, then base angles are congruent. If then

Converse of Isosceles Triangle Conjecture If a triangle has two congruent angles, then it is an isosceles triangle. If then

Equilateral Triangle Conjecture An equilateral triangle is equiangular, and an equiangular triangle is equilateral.

Find the missing angle measures. <68° and < a are base angles  therefore they are congruent b ma = 68˚ Triangle sum to find <b m<b = 180 – 68 - 68 m<b = 180 -136 mb = 44˚ 68˚ a

<c & <d are base angles and are congruent Find the missing angle measures. Triangle sum = 180° 180 = 119 + c + d 180 – 119 = c + d 61 = c + d <c & <d are base angles and are congruent <c = ½ (61) = 30.5 <d = ½ (61) = 30.5 mc = md = 30.5˚ 119˚ 30.5˚ c d

EFG is an equilateral triangle Therefore <E = <F = <G Find the missing angle measures. E EFG is an equilateral triangle Therefore <E = <F = <G 180 /3 = 60 mE = mF = mG = 60˚ 60˚ G F 60˚

Find the missing angle measures. Find mG. ∆GHJ is isosceles < G = < J x + 44 = 3x 44 = 2x x = 22 Thus m<G = 22 + 44 = 66° And m<J = 3(22) = 66°

Find the missing angle measures. Find mN Base angles are = 6y = 8y – 16 -2y = -16 y= 8 Thus m<N = 6(8) = 48°. m<P = 8(8) – 16 = 48°

Find the missing angle measures. Using Properties of Equilateral Triangles Find the value of x. ∆LKM is equilateral. m<K = m<L = m<M 180/3 = 60° 2x + 23 = 60 2x = 37 x = 18.5°

Find the missing side measures. Using Properties of Equilateral Triangles Find the value of y. ∆NPO is equiangular therefore, ∆NPO is also equilateral. ft ft 5y – 6 = 4y +12 y – 6 = 12 y = 18 Side NO = 5(18) – 6 = 90ft

Find the missing angle measures. Using the symbols describing shapes answer the following questions: 36o a b c 45o d Isosceles triangle Two angles are equal Equilateral triangle all angles are equal Right-angled triangle a = 36o c = 180 ÷ 3 = 60o d = 180 – (45 + 90) = 45o b = 180 – (2 × 36) = 108o

Find the missing angle measures. Kite - Made up of 2 isosceles triangles q 36o p s r 56o p = 36o q = 180 – (2 * 36) = 108o 56 + (r + s) = 180o (r + s) = 180 – 56 = 124 Because r = s r = s = 124 ÷ 2 = 62o

Find the missing angle measures. a = 64o Equilateral triangle b = 180 – (2 ×64o ) = 52o h = i c = d e = f = g = 60o h + i = 180 - 90 c + d = 180 - 72 h + i = 90 c + d = 108 h = i = 45o c = d = 54o

p = 50o q = 180 – (2 ×50o ) = 80o r = q = 80o vertical angles are equal Therefore : s = t = p = 50o

Find the missing angle measures. Properties of Triangles p = q = r = 60o a = b= c = 60o d = 180 – 60 = 120o s = t = 180 - 43 = 68.5o 2 e + 18 = a = 60 exterior angle = sum of remote interior angles e = 60 – 18 = 42o

Find the missing angle measures. z Find the value of x Find the value of y x is a base angle 180 = x + x + 50 130 = 2x x = 65° 2) y & z are remote interior angles and base angles of an isosceles triangle Therefore: y + z = x and y = z y + z = 80° y = 40°

Find the missing angle measures. Find the value of x Find the value of y 1) ∆CDE is equilateral All angles = 60° Using Linear Pair <BCD = 70° x is the vertex angle x = 180 – 70 – 70 x = 40° 70° 60° 2) y is the vertex angle y = 180 – 100 y = 80°

Find the missing measures. Lesson Quiz Find each angle measure. 1. mR 2. mP Find each value. 3. x 4. y 5. x

Find the missing angle measures. Lesson Quiz con’t 6. The vertex angle of an isosceles triangle measures (a + 15)°, and one of the base angles measures 7a°. Find a and each angle measure. (Make a sketch)

Lesson Quiz Solutions 28° 124° 20 6 26 a = 11; 26°; 77°; 77°

Homework In your textbook: Lesson 4.2/ 1-11