4.10. Isosceles Triangles http://www.ck12.org
The base angles should each be 72. 5 ◦and the vertex angle should be 35 ◦.
We can generalize this investigation into the Base Angles Theorem.
Base Angles Theorem:The base angles of an isosceles triangle are congruent.
To prove the Base Angles Theorem, we will construct the angle bisector through the vertex angle of an isosceles
triangle.
Given: Isosceles triangle 4 DEFwithDE∼=EF
Prove:^6 D∼=^6 F
TABLE4.18:
Statement Reason- Isosceles triangle 4 DEFwithDE∼=EF Given
- Construct angle bisectorEGfor^6 E Every angle has one angle bisector
3.^6 DEG∼=^6 F EG Definition of an angle bisector
4.EG∼=EG Reflexive PoC - 4 DEG∼= 4 F EG SAS
6.^6 D∼=^6 F CPCTC
By constructing the angle bisector,EG, we designed two congruent triangles and then used CPCTC to show that the
base angles are congruent. Now that we have proven the Base Angles Theorem, you do not have to construct the
angle bisector every time. It can now be assumed that base angles of any isosceles triangle are always equal. Let’s
further analyze the picture from step 2 of our proof.
Because 4 DEG∼= 4 F EG, we know that^6 EGD∼=^6 EGFby CPCTC. Thes two angles are also a linear pair, so
they are congruent supplements, or 90◦each. Therefore,EG⊥DF. Additionally,DG∼=GFby CPCTC, soGis the
midpoint ofDF. This means thatEGis theperpendicular bisectorofDF, in addition to being the angle bisector
of^6 DEF.
Isosceles Triangle Theorem:The angle bisector of the vertex angle in an isosceles triangle is also the perpendicular
bisector to the base.
The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true.