HL with a Shared Leg
Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 30 of 63.
A common HL figure is one triangle cut in two by a line at right angles. The cutting line is a leg of both halves, so the Reflexive property gives the leg pair.
| Statement | Reason |
|---|---|
| 1. PQ = PS | Given |
| 2. ∠PQR is a right angle | Given (the square) |
| 3. ∠PSR is a right angle | Given (the square) |
| 4. PR = PR | Reflexive property |
| 5. △PQR ≅ △PSR | HL, using lines 1 to 4 |
Here the shared side is the hypotenuse instead, opposite both right angles. Either way, HL needs one pair of hypotenuses and one pair of legs, and a shared side can be either one.
Challenge
In △ABC the ticks show BA = BC, and the height BD meets AC at right angles. Prove that △ABD ≅ △CBD using HL, then press Check.
Try it yourself
Write the proof
Test yourselfThis 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 ProveStatements and ReasonsChaining Equal PartsYour First Full ProofRecap: A Chain of Reasons2Congruent Triangles
Same Size, Same ShapeMatching SidesMatching AnglesWhat Congruence GivesRecap: Matching Parts8Proofs with Vertical Angles
Vertical Angles in ProofsThe BowtieBowtie with ASALinear Pairs in ProofsRecap: Vertical Angles