Examples of congruent triangles HL

📚 Quick Study Guide

• 📐 Definition: The Hypotenuse-Leg (HL) Theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
• 📝 Conditions: This theorem applies *only* to right triangles. You must confirm that both triangles have a 90-degree angle.
• 📏 Hypotenuse: The hypotenuse is the side opposite the right angle. It's also the longest side of the right triangle.
• 🦵 Legs: The legs are the two sides that form the right angle.
• ✅ Congruence: To use HL, you need to show that the hypotenuses are congruent and *one* pair of corresponding legs are congruent.
• 💡 Symbolic Representation: If $\triangle ABC$ and $\triangle DEF$ are right triangles, $\angle B$ and $\angle E$ are right angles, $\overline{AC} \cong \overline{DF}$ (hypotenuses), and $\overline{AB} \cong \overline{DE}$ (legs), then $\triangle ABC \cong \triangle DEF$ by HL.

Practice Quiz

1. Which of the following conditions is necessary to apply the Hypotenuse-Leg (HL) Theorem?

   1. Both triangles must be equilateral.
   2. Both triangles must be right triangles.
   3. Both triangles must be isosceles.
   4. Both triangles must be acute.

2. In right triangles $\triangle PQR$ and $\triangle XYZ$, $\overline{PR}$ and $\overline{XZ}$ are the hypotenuses. Which additional piece of information is needed to prove $\triangle PQR \cong \triangle XYZ$ by HL?

   1. $\overline{PQ} \cong \overline{XY}$
   2. $\angle Q \cong \angle Y$
   3. $\overline{QR} \cong \overline{PR}$
   4. $\angle P \cong \angle X$

3. Given right triangles $\triangle ABC$ and $\triangle DEF$ with right angles at B and E respectively. If $\overline{AC} \cong \overline{DF}$, which additional congruence is sufficient to prove $\triangle ABC \cong \triangle DEF$ by the HL Theorem?

   1. $\overline{AB} \cong \overline{DE}$
   2. $\angle A \cong \angle D$
   3. $\angle C \cong \angle F$
   4. $\overline{BC} \cong \overline{AC}$

4. Two right triangles have hypotenuses of length 10 cm. One leg of the first triangle is 6 cm, and one leg of the second triangle is also 6 cm. Can you conclude that the two triangles are congruent using HL?

   1. Yes, they are congruent by HL.
   2. No, you need more information.
   3. Only if the angles are also congruent.
   4. Only if the triangles are similar.

5. In $\triangle LMN$, $\angle M$ is a right angle, and in $\triangle OPQ$, $\angle P$ is a right angle. If $\overline{LN} \cong \overline{OQ}$ and $\overline{LM} \cong \overline{OP}$, which theorem proves $\triangle LMN \cong \triangle OPQ$?

   1. SSS
   2. SAS
   3. ASA
   4. HL

6. Right triangles $\triangle STU$ and $\triangle VWX$ have hypotenuses $\overline{SU}$ and $\overline{VX}$ respectively. If $\overline{SU} \cong \overline{VX}$ and $\overline{ST} \cong \overline{VW}$, what can be concluded?

   1. $\triangle STU \cong \triangle VWX$ by ASA
   2. $\triangle STU \cong \triangle VWX$ by HL
   3. $\triangle STU \cong \triangle VWX$ by SAS
   4. No congruence can be determined.

7. Given two right triangles, $\triangle ABC$ and $\triangle DBC$, sharing a common leg $\overline{BC}$. If the hypotenuses $\overline{AC}$ and $\overline{DB}$ are congruent, what justifies the congruence of the triangles?

   1. SAS
   2. SSS
   3. HL
   4. ASA