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SAS with a Shared Side

Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 18 of 63.

A shared side can be one of the S letters in SAS. In the figure, △PQR and △PQS sit on either side of PQ. The ticks give QR = QS and the arcs give ∠PQR = ∠PQS.

$Triangles PQR and PQS sharing PQ, with QR and QS ticked and the angles at Q marked$

Statement	Reason
1. QR = QS	Given
2. ∠PQR = ∠PQS	Given
3. PQ = PQ	Reflexive property
4. △PQR ≅ △PQS	SAS, using lines 1, 2 and 3

Check that the angle is included: in △PQR the sides QR and PQ meet at Q, and the marked angle is at Q. The matching keeps P and Q fixed, and sends R to S.

Challenge

In △ABC the ticks show BA = BC, and BD splits the angle at B into two angles marked equal. Prove that △ABD ≅ △CBD, then press Check.

Try it yourself

$Write the proof$

Write the proof

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Congruence & Proof

1What a Proof Is

Given and Prove Statements and Reasons Chaining Equal Parts Your First Full Proof Recap: A Chain of Reasons

4SAS

The Included Angle The SAS RuleSAS with a Shared SideWhy SSA Fails SAS or SSS?Recap: SAS

7Using CPCTC

Congruent, Then Equal Proving a Bisector Flowchart Proofs Proving a Midpoint Recap: Using CPCTC

10Isosceles Triangle Proofs

Base Angles in a Proof Equal Angles, Equal Sides Proving Isosceles The Perpendicular Bisector Chasing in Isosceles Shapes Recap: Isosceles Proofs

2Congruent Triangles

Same Size, Same Shape Matching Sides Matching Angles 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 Bowtie Bowtie with ASA Linear Pairs in Proofs Recap: Vertical Angles

11Similarity Proofs

AA Similarity Two Right Triangles SSS Similarity SAS Similarity Similar, Then Find Recap: Similarity