Bowman’s Website

October 30, 2009

Geometry Notes — Proving Triangles Congruent 4 — Right Triangles — HL, HA, LL, LA

Filed under: Geometry, Geometry Notes — Tags: — bowman @ 7:13 am

If you have the right three pairs of corresponding congruent parts from two triangles, that is all you need to prove that two triangles are congruent.

If the two triangles you are working with are right triangles, you already have one of those three parts.

To prove that two right triangles are congruent, consider these combinations

HL – Hypotenuse Leg
HA – Hypotenuse Acute angle
LL – Leg Leg
LA – Leg Acute angle

Of these four, only HL needs to mastered.

Why?

1. The Hypotenuse-Acute Angle Theorem is a rule specially designed for use with right triangles. It states if the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, the two triangles are congruent.

Look at the two right triangles and the congruence marks that are given. Before the hypotenuse-acute angle theorem was ever mentioned, you already had the means to declare these two triangles congruent. You would have said the two triangles were congruent because of Angle-Angle-Side.

2. The Leg-Leg Theorem is a rule specially designed for use with right triangles. It states if the legs of one right triangle are congruent to the legs of another right triangle, the two right triangles are congruent.

Again, look at the two right triangles and the congruence marks that are given. You would have said the two triangles were congruent because of Side-Angle-Side.

3. The Leg-Acute Angle Theorem is a rule specially designed for use with right triangles. It states if a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.

Look at the two right triangles and the congruence marks that are given. Before you were lucky today by being around me today, you already had the wonderful knowledge of Angle-Side-Angle.

So instead of learning more about HA, LL, and LA, we will continue to use the rules AAS, SAS, and ASA. These last three are more useful because they work for any triangle. HA, LL, and LA are right triangle specific.

The Hypotenuse Leg Theorem

Hypotenuse Leg Theorem
(HL)  If a leg and the hypotenuse of one right triangle are congruent to a leg and the hypotenuse of a second right triangle, then the two right triangles are congruent.

This is the one case where SSA is valid. The angle must be a right angle, thus the triangles must be right triangles.

Since ΔBOW and ΔMAN are right triangle,  BOMA and OWAN , then ΔBOW ≅ ΔMAN by HL.

Congruent Triangles – Hypotenuse and leg of a right triangle (HL): Two right triangles are congruent if the hypotenuse and one corresponding leg are congruent in both triangles.

A combination of two lengths – the hypotenuse and a leg – are highlighted to indicate they are the parts being used to test for congruence.

Here’s a reason why this works. Since we know the hypotenuse and one other side, the third side can be accurately determined, due to the Pythagorean Theorem.  So this is really a version of the SSS case. (side-side-side).

What does this mean? Because the triangles are congruent:
1. The remaining third sides are congruent (PQLM)
2. The other two angles are congruent (∠R ≅ ∠N and ∠Q ≅ ∠M)

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