5.4 Isosceles and Equilateral Triangles Geometry What conjectures can you make about congruent angles and sides?
Geometry 5.4 Isosceles, Equilateral Triangles Topic/Objective Use properties of isosceles triangles. Use properties of equilateral triangles. April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Opposite Angles and Sides EF is opposite D. E is opposite side DF. D F April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Isosceles Triangles Vertex Angle Leg Leg Base Angles Base April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles A construction. Begin with an isosceles triangle, ABC. Draw the angle bisector from the vertex angle. The angle bisector intersects the base at M. ACM BCM. Why? SAS A B. Why? CPCTC C A B M April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Theorem 5.6 Base Angles Theorem. If two sides of a triangle are congruent, then the angles opposite them are congruent. (Easy form) The base angles of an isosceles triangle are congruent. April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Visually: This: Means this: The base angles of an isosceles triangle are congruent. April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Example Problem Solve for x. x + x + 52 = 180 2x + 52 = 180 2x = 128 x = 64 52° x° x° April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Example 2 Solve for x and y. In an isosceles triangle, base angles are congruent. So y is… 42° Now use the triangle angle sum theorem: x + 42 + 42 = 180 x + 84 = 180 x = 96° 42° x° y° April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Example 3. You try it. 50° x° y° x = 65° y = 32.5° Find x and y. 50° y° 2y + 115 = 180 2y = 65 y = 32.5° y° 32.5°` 115° x° 65° 65° x° 2x + 50 = 180 2x = 130 x = 65 180 – 65 = 115 April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Example 4 Solve for x. (2x)° (3x – 25)° 3x – 25 = 2x x = 25 April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Theorem 5.7 Converse of the Base Angles Theorem. If two angles of a triangle are congruent, then the sides opposite them are congruent. April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Example 5 Since base angles are equal, opposite sides are equal. 4x + 52 = 2x + 68 2x + 52 = 68 2x = 16 x = 8 4x + 52 2x + 68 4(8) + 52 = 84 Solve for x, then find the length of the legs. April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Example 6 You do it. Find the length of each side. 5x = 3x + 16 2x = 16 x = 8 40 4x – 2 5x 30 3x + 16 40 April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Equilateral Triangles Corollaries to Base Angles Theorem If a triangle is equilateral, then it is also equiangular. If a triangle is equiangular, then it is also equilateral. April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Example 7 Solve for x. All sides are congruent. 3x – 10 = x + 10 2x = 20 x = 10 3x – 10 x + 10 2x 2x = x + 10 x = 10 3x – 10 = 2x x – 10 = 0 x = 10 April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles One last problem. Solve for x and y. x° y° 50° Solution… April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Solution… This triangle is equilateral. Each angle is? 40° x° 70° 80° y° 60° 70° ? 50° 50° 60° 60° These angles form straight angle. The missing angle is? April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles
Geometry 5.4 Isosceles, Equilateral Triangles Summarize what you have learned today The base angles of an isosceles triangle are congruent. Equilateral triangles are Equiangular. April 25, 2018 Geometry 5.4 Isosceles, Equilateral Triangles