Bowtie with ASA

Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 39 of 63.

Arcs and ticks can be spread out in a bowtie. Then the vertical angles pair with whatever else is marked, and the rule depends on where the side sits.

$Bowtie with BC and CE ticked, and arcs at B and E$

Statement	Reason
1. ∠CBA = ∠CED	Given
2. ∠BCA = ∠ECD	Vertical angles
3. BC = EC	Given
4. △ABC ≅ △DEC	ASA, using lines 1, 3 and 2

Here the two angle pairs are at B and C, and the ticked side BC joins B and C: it sits between the two angles, so the rule is ASA. Compare: if the ticks were on the outer sides AB and DE instead, the side would no longer join the two angles, and the same angles would prove it by AAS. Before you name the rule, find the vertices of the two angle pairs and check whether the ticked side joins them.