Using the Isosceles Triangle Theorems to Solve Proofs
Using the Angle-Side-Angle Method to Prove Triangles Congruent
Using the Side-Side-Side Method to Prove Triangles Congruent

Using the Hypotenuse-Leg-Right Angle Method to Prove Triangles Congruent

The HLR (Hypotenuse-Leg-Right angle) theorem — often called the HL theorem — states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. The following figure shows you an example.

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You can call this theorem HLR (instead of HL) because its three letters emphasize that before you can use it in a proof, you need to have three things in the statement column (congruent hypotenuses, congruent legs, and right angles).

Note: When you use HLR, listing the pair of right angles in a proof statement is sufficient for that part of the theorem; you don’t need to state that the two right angles are congruent.

Ready for an HLR proof? Well, ready or not, here you go.

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Here’s a possible game plan. You see the pair of congruent triangles and then ask yourself how you can prove them congruent. You know you have a pair of congruent sides because the triangle is isosceles.

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Here’s the proof:

Statement 1:

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Reason for statement 1: Given.

Statement 2:

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Reason for statement 2: Definition of isosceles triangle.

Statement 3:

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Reason for statement 3: Reflexive Property.

Statement 4:

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Reason for statement 4: Given.

Statement 5:

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Reason for statement 5: Definition of altitude.

Statement 6:

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Reason for statement 6: Definition of perpendicular.

Statement 7:

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Reason for statement 7: HLR (using lines 2, 3, and 6)

Statement 8:

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Reason for statement 8: CPCTC.

Statement 9:

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Reason for statement 9: Definition of midpoint.

Statement 10:

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Reason for statement 10: Definition of median.

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How to Do an Indirect Proof
Using Two Equidistant Points to Determine a Perpendicular Bisector
Using a Point on a Perpendicular Bisector to Prove two Segments Congruent
Using the Side-Angle-Side Method to Prove Triangles Congruent
Using the Angle-Angle-Side Method to Prove Triangles Congruent
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