Proving a Bisector

Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 33 of 63.

A ray bisects an angle when it cuts it into two equal halves. The definition works both ways, and both ways are a proof reason, the Definition of angle bisector. If a ray bisects an angle, the halves are equal. If the halves are equal, the ray bisects the angle.

$Kite KLMN with ticks on KL, KN and ML, MN, and the diagonal KM$

Statement	Reason
1. KL = KN	Given
2. ML = MN	Given
3. KM = KM	Reflexive property
4. △KLM ≅ △KNM	SSS, using lines 1, 2 and 3
5. ∠LKM = ∠NKM	CPCTC, using line 4
6. KM bisects ∠LKN	Definition of angle bisector, using line 5

The definition only turns equal halves into a bisector, so the hard part is the halves. CPCTC got them here. On the board, a bisector statement is written by tapping the two halves of the angle.