4.5. Isosceles and Equilateral Triangles http://www.ck12.org
47 ◦+ 47 ◦+v= 180 ◦
v= 180 ◦− 47 ◦− 47 ◦
v= 86 ◦Example 3:If an isosceles triangle has a vertex angle with a measure of 116◦, what is the measure of each base
angle?
Solution:Draw a picture and set up and equation to solve for the base angles,b. Recall that the base angles are
equal.
116 ◦+b+b= 180 ◦
2 b= 64 ◦
b= 32 ◦Example 4:Algebra ConnectionFind the value ofxand the measure of each angle.
Solution:Set the angles equal to each other and solve forx.
( 4 x+ 12 )◦= ( 5 x− 3 )◦
15 ◦=xIfx= 15 ◦, then the base angles are 4( 15 ◦)+ 12 ◦, or 72◦. The vertex angle is 180◦− 72 ◦− 72 ◦= 36 ◦.
The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true.
Base Angles Theorem Converse: If two angles in a triangle are congruent, then the opposite sides are also
congruent.
So, for a triangle 4 ABC, if^6 A∼=^6 B, thenCB∼=CA.^6 Cwould be the vertex angle.
Isosceles Triangle Theorem Converse:The perpendicular bisector of the base of an isosceles triangle is also the
angle bisector of the vertex angle.
In other words, if 4 ABCis isosceles,AD⊥CBandCD∼=DB, then^6 CAD∼=^6 BAD.