Proving Isosceles

Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 46 of 60.

Often the equal angles are not inside the triangle. They may be outside, on lines that extend the sides, and you have to bring them in first. Here linear pairs and supplements do it; where lines cross, vertical angles can do it too.

$Triangle ABC with the base extended to D and E; the outside angles at B and C are marked equal$

Statement	Reason
1. ∠ABD = ∠ACE	Given
2. ∠ABD and ∠ABC are supplementary	Linear pair
3. ∠ACE and ∠ACB are supplementary	Linear pair
4. ∠ABC = ∠ACB	Supplements of equal angles, using lines 1, 2 and 3
5. AB = AC	Sides opposite equal angles, using line 4

The outside angles are equal, so the inside angles, 180° minus each, are equal. Then the converse turns the equal angles into equal sides.

Challenge

The base VW of △XVW lies on the line DE, and the sides XV and XW are extended past the base to U and Y. The angles ∠DVU and ∠EWY are marked equal. Prove that XV = XW, then press Check.

Try it yourself

$Write the proof$