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SSS with a Shared Side

Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 14 of 63.

Put the last two lessons together. In the kite below, the ticks give two pairs of sides, and the diagonal is a side of both triangles. Three pairs: SSS.

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

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

Check the letter order before you write the congruence. K matches K and M matches M, because the shared side is KM in both. That leaves L to match N, and indeed KL = KN and ML = MN.

Challenge

In quadrilateral ABCD the ticks show AB = CD and BC = DA, and the diagonal AC is drawn. Prove that △ABC ≅ △CDA, 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

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

3SSS

Three Sides Fix It The SSS Rule A Shared SideSSS with a Shared SideRecap: SSS

6HL for Right Triangles

Hypotenuse and Legs The HL Rule Why HL Works HL with a Shared Leg Recap: HL

9Proofs with Parallel Lines

Parallel Angles as Reasons Alternate Angles in a Proof Corresponding Angles in Proofs Parallel or Not?Recap: Parallel Proofs

12Final Challenges

Final: The Kite Diagonals Final: Overlapping Triangles Final: Enough to Find It?Final: The Height Bisects Final: The Roof Frame