Bowtie with ASA

Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 36 of 60.

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.

Challenge

Segments TW and UX cross at V. The ticks show TV = WV, and the angles at T and W are marked equal. Prove that △TUV ≅ △WXV, then press Check.

Try it yourself

$Write the proof$

Write the proof

All lessons in Congruence & Proof

2Congruent Triangles

Same Size, Same Shape What Congruence Gives Recap: Matching Parts

5ASA and AAS

The Side Between The ASA Rule The AAS Rule ASA or AAS?Recap: ASA and AAS

8Proofs with Vertical Angles

Vertical Angles in Proofs The BowtieBowtie with ASALinear Pairs in Proofs Recap: Vertical Angles

11Similarity Proofs