4.2 Congruence in right triangles

Congruence in Right Triangles

Right triangles have a built-in piece of information that other triangles don't: the right angle. Because you already know one angle is 90°, you need less additional information to prove two right triangles congruent. This section covers the congruence theorems specific to right triangles, how the Pythagorean theorem supports congruence proofs, and the side ratios in special right triangles.

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Congruence Theorems for Right Triangles

Hypotenuse-Leg (HL) Theorem

The HL theorem states: if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and the corresponding leg of another right triangle, then the two triangles are congruent.

• Example: If $\overline{AC} \cong \overline{DF}$ (hypotenuses) and $\overline{AB} \cong \overline{DE}$ (legs), then $\triangle ABC \cong \triangle DEF$.

This theorem only works for right triangles. It's actually a special case of SSA (side-side-angle), which normally does not prove congruence for general triangles. The right angle constrains the geometry enough that the ambiguous case can't occur.

To apply HL in a proof:

1. Confirm both triangles are right triangles (look for right angle marks or given information).
2. Identify the hypotenuse and one leg in each triangle.
3. Show the hypotenuses are congruent and the corresponding legs are congruent.
4. Conclude the triangles are congruent by HL.

Leg-Leg (LL) Theorem

The LL theorem states: if both legs of one right triangle are congruent to the corresponding legs of another right triangle, then the two triangles are congruent.

• Example: If $\overline{AB} \cong \overline{DE}$ and $\overline{BC} \cong \overline{EF}$, then $\triangle ABC \cong \triangle DEF$.

This is really just SAS applied to right triangles. The two legs are the sides, and the included angle is the right angle (which is congruent in both triangles by definition). You'll sometimes see this written as LL in right-triangle proofs, but citing SAS works too.

To apply LL:

1. Confirm both triangles are right triangles.
2. Identify the two legs in each triangle.
3. Show the corresponding legs are congruent.
4. Conclude the triangles are congruent by LL (or SAS).

Using the Pythagorean Theorem in Congruence Problems

The Pythagorean theorem states that in a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse and $a$ and $b$ are the legs.

This theorem doesn't prove congruence on its own, but it fills in missing side lengths so you can then apply a congruence theorem. For instance, if you know the hypotenuse is 13 and one leg is 5, you can find the other leg:

$5^2 + b^2 = 13^2$

$b^2 = 169 - 25 = 144$

$b = 12$

Once you've found that missing length, you might have enough information to use HL, LL, or SSS to prove congruence.

Steps for using the Pythagorean theorem in a congruence proof:

1. Identify the right triangles in the problem.
2. Use $a^2 + b^2 = c^2$ to calculate any missing side lengths.
3. With all necessary sides known, apply the appropriate congruence theorem (HL, LL, SSS, SAS, or ASA).

Properties of Special Right Triangles

45-45-90 Triangles

A 45-45-90 triangle is an isosceles right triangle: the two legs are congruent, and the angles measure 45°, 45°, and 90°.

The side ratio is $x : x : x\sqrt{2}$.

• If a leg has length $x$, the hypotenuse has length $x\sqrt{2}$.
• Working backward: if the hypotenuse has length $h$, each leg has length $\frac{h}{\sqrt{2}} = \frac{h\sqrt{2}}{2}$.

For example, if a leg is 7, the hypotenuse is $7\sqrt{2}$. If the hypotenuse is 10, each leg is $\frac{10\sqrt{2}}{2} = 5\sqrt{2}$.

30-60-90 Triangles

A 30-60-90 triangle has angles of 30°, 60°, and 90°. You can think of it as half of an equilateral triangle split down the middle.

The side ratio is $x : x\sqrt{3} : 2x$.

• The short leg (opposite 30°) has length $x$.
• The long leg (opposite 60°) has length $x\sqrt{3}$.
• The hypotenuse (opposite 90°) has length $2x$.

For example, if the short leg is 4, the long leg is $4\sqrt{3}$ and the hypotenuse is 8.

To use special right triangle properties in a problem:

1. Identify the triangle type from the given angles or from a side ratio that matches one of the patterns above.
2. Set the known side equal to the appropriate part of the ratio and solve for $x$.
3. Use $x$ to find the remaining sides.
4. If the problem asks for congruence, apply the appropriate theorem once you have the side lengths.