Properties of Isosceles & Equilateral Triangles Lesson 51 Properties of Isosceles & Equilateral Triangles

Review Vocabulary An equilateral triangle has all congruent sides ΔFGH An equiangular triangle has all congruent angles ΔABC An isosceles triangle has at least 2 congruent sides ΔJIK

New Vocabulary A leg of an isosceles triangle is one of the two congruent sides 𝑆𝑇 & 𝑅𝑇 The vertex angle of an isosceles triangle is the angle formed by the legs ∠T The base of an isosceles triangle is the side opposite the vertex angle 𝑅𝑆 A base angle of an isosceles triangle is one of the two angles that have the base as a side (not the vertex angle) ∠S & ∠R

Theorem 51-1: Isosceles Triangle Theorem If a triangle is isosceles, then its base angles are congruent. ∆𝑅𝑆𝑇 is isosceles Therefore, ∠𝑆≅∠𝑅

Can you apply Theorem 51-1 to equilateral triangles? Why? Yes, because equilateral triangles are also isosceles What can you conclude about all three angles of an equilateral triangle? Explain. They are all congruent ∠G & ∠H are base angles so ∠G ≅ ∠H ∠F & ∠H are also base angles so ∠F ≅ ∠H Therefore, using the transitive property all three angles are congruent

Theorem 51-2: Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are also congruent. In ∆𝑅𝑆𝑇 ∠𝑆≅∠𝑅 Therefore, 𝑆𝑇 ≅ 𝑅𝑇

Can you apply Theorem 51-2 to equiangular triangles? Yes What can you conclude about all three sides of an equiangular triangle? Why? They are all congruent In ΔABC, ∠B ≅ ∠C ≅ ∠A Therefore, 𝐴𝐶 ≅ 𝐵𝐴 ≅ 𝐶𝐵

Corollaries of Equilateral & Equiangular Triangles Corollary 51-1-1 Corollary 51-2-1 If a triangle is equilateral, then it is equiangular. If a triangle is equiangular, then it is equilateral.

Theorem 51-3 If a line bisects the vertex angle of an isosceles triangle, then it is the perpendicular bisector of the base. ΔSTR is isosceles and 𝑇𝑈 bisects ∠STR Therefore, ∠TUS is a right angle and 𝑆𝑈 ≅ 𝑈𝑅

Theorem 51-4 If a line is the perpendicular bisector of the base of an isosceles triangle, then it bisects the vertex angle. ΔSTR is isosceles and 𝑇𝑈 is the perpendicular bisector of 𝑆𝑅 Therefore, ∠STU ≅ ∠RTU

Δ𝐼𝐽𝐾 is isosceles & ∠𝐼 is the vertex angle Find the 𝑚∠𝐼 & 𝑚∠𝐾 𝑚∠𝐾=28° 𝑚∠𝐼=180− 28+28 𝑚∠𝐼=124°

The perimeter of ΔABC is 21 in. & ∠A ≅ ∠B Find the length of 𝐴𝐶 if 𝐴𝐵=10 𝑖𝑛. If ∠A ≅ ∠B, then which sides are congruent? 𝐴𝐶 ≅ 𝐵𝐶 𝐴𝐶+𝐵𝐶+𝐴𝐵=21 2 𝐴𝐶 +10=21 2 𝐴𝐶 =11 𝐴𝐶=5 1 2 𝑖𝑛.

Looking Forward Applying properties of Isosceles and Equilateral Triangles will prepare you for: Lesson 55: Triangle Midsegment Theorem Lesson 69: Properties of Trapezoids and Kites Lesson 85: Cross Sections of Solids