Question

Prove that the line joining the point of intersection of two angular bisectors of the base angles of an isosceles triangle to the vertex bisects the vertical angle.

~Thanks in advance ! ツ​

Answer 1

See Below.

Explanation:

Please refer to the attachment below.

Essentially, we want to prove that AE bisects ∠A.

Statements: Reasons:

`1)\text{ } \Delta ABC \text{ is isosceles}` Given

`2)\text{ } m\angle C=m\angle B` Isosceles Triangle Theorem

`3)\text{ }m\angle C=m\angle ACE+m\angle ECD` Angle Addition

`4)\text{ } CE\text{ bisects }\angle C` Given

`5)\text{ } m\angle ACE=m\angle ECD` Definition of Bisector

`6)\text{ } m\angle C=2m\angle ECD` Substitution

`7)\text{ } m\angle B=\angle ABE+m\angle EBD` Angle Addition

`8)\text{ } BE\text{ bisects } \angle B` Given

`9)\text{ }m\angle ABE=m\angle EBD` Definition of Bisector

`10)\text{ } m\angle B=2m\angle EBD` Substitution

`11)\text{ } 2m\angle ECD=m\angle EBD` Substitution

`12)\text{ } m\angle ECD=m\angle EBD` Division Property of Equality

`13)\text{ } CE=BE` Isosceles Triangle Theorem

`14)\text{ } AC=AB` Isosceles Triangle Theorem

`15)\text{ } AE=AE` Reflexive Property

`16)\text{ } \Delta AEC\cong \Delta AEB` SSS Congruence

`17)\text{ } \angle CAE\cong \angle BAE` CPCTC

`18)\text{ } AE\text{ is a bisector of } \angle A` Converse of Bisector