Download Chapter 4 - Congruent Triangles

Unit 5 – Congruent Triangles
Lesson 5 - Isosceles and Equilateral Triangles (p. 210)
The Isosceles Triangle Theorems
 Legs of an isosceles triangle - __________________________________________________
 Base of an isosceles triangle - __________________________________________________
 Vertex angle of an isosceles triangle - ___________________________________________
 Base angles of an isosceles triangle - ____________________________________________
Theorem 4-3 (Isosceles Triangle Theorem) – If two sides of a triangle are congruent, then the
angles opposite those sides are congruent.
A  B
Theorem 4-4 (Converse of Isosceles Triangle Theorem) – If two angles of a triangle are
congruent, then the sides opposite the angles are congruent.
AC  BC
Theorem 4-5 – The bisector of the vertex angle of an isosceles triangle is the perpendicular
bisector of the base.
1. CD bisects ACB (cuts the vertex angle into two congruent angles)
2. CD  AB (creates right angles with the base)
3. CD bisects AB (cuts the base into two congruent segments)
- Proving the Isosceles Triangle Theorem
a. Developing Proof To prove the Isosceles Triangle Theorem,
begin with isosceles XYZ with XY  XZ . Draw XB , the
bisector of the vertex angle YXZ .
Given: XY  XZ , XB bisects YXZ .
Prove: Y  Z
Lesson 5 – Isosceles and Equilateral Triangles
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Revised Fair 2014-2015
Unit 5 – Congruent Triangles
- Using the Isosceles Triangle Theorem
a. Developing Proof Explain why RST is isosceles. Explain.
b. Can you conclude that RUV is isosceles? Explain.
- Using Algebra
a. Find the values of x and y.
b. Suppose mL  43 . Find the values of x and y.
 Corollary - ________________________________________________________________
Corollary to Theorem 4-3 – If a triangle is equilateral, then the triangle is equiangular.
X  Y  Z
Corollary to Theorem 4-4 – If a triangle is equiangular, then the triangle is equilateral.
XY  YZ  ZX
- Real-World Connection
a. Landscaping A landscaper uses rectangles and equilateral triangles for the path
around the hexagonal garden. Find the value of x.
b. What is the measure of the angle at each outside corner of the path?
Lesson 5 – Isosceles and Equilateral Triangles
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Revised Fair 2014-2015