Advanced Applications of the HL Theorem

The final page presents a sophisticated proof that combines the Hypotenuse-Leg Theorem with the concept of perpendicular bisectors.

Given:

• CD ≅ EA
• AD is the perpendicular bisector of CE

Prove: ΔCBD ≅ ΔEBA

The proof systematically uses the given information and the properties of perpendicular bisectors to establish the conditions necessary for applying the HL Theorem.

Vocabulary: A perpendicular bisector is a line that passes through the midpoint of another line segment at a right angle.

The proof demonstrates how to:

1. Identify the hypotenuse (CD ≅ EA)
2. Use the properties of perpendicular bisectors to establish leg congruence (CB ≅ EB)
3. Confirm right angles using the definition of perpendicular lines
4. Apply the HL Theorem to conclude triangle congruence

Highlight: This advanced proof showcases how the HL Theorem can be applied in complex geometric scenarios, particularly when combined with properties of perpendicular bisectors.

This example reinforces the versatility and power of the Hypotenuse-Leg Theorem in proving congruence for right triangles, even in more intricate geometric configurations.

Congruence in Right Triangles: Hypotenuse-Leg (HL) Theorem

The Hypotenuse-Leg (HL) Theorem is introduced as a fundamental concept for proving congruence in right triangles. This 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.

Definition: The Hypotenuse-Leg 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 page provides a visual representation of the theorem using triangles PQR and XYZ, demonstrating how the theorem is applied when the hypotenuse and one leg of each triangle are congruent.

Example: Given right triangles PQR and XYZ, if PR ≅ XZ (hypotenuse) and PQ ≅ XY (leg), then ΔPQR ≅ ΔXYZ.

An example proof is presented to demonstrate the application of the HL Theorem:

Given:

• ∠PRS and ∠RPQ are right angles
• SP ≅ OR

Prove: ΔPRS ≅ ΔRPQ

The proof proceeds step-by-step, using the definition of right angles and the given information to apply the HL Theorem and conclude that the triangles are congruent.

Highlight: The HL Theorem is a powerful tool for proving congruence in right triangles when the hypotenuse and one leg of each triangle are known to be congruent.

Applying the HL Theorem in Proofs

This page expands on the application of the Hypotenuse-Leg Theorem by presenting more complex proofs and introducing related concepts.

The first example builds on the previous proof, introducing additional given information:

• PS ≅ PT
• PRS ≅ PRT

The goal is to prove that ΔPRS ≅ ΔPRT. The proof utilizes the Isosceles Triangle Theorem in conjunction with the HL Theorem to establish congruence.

Vocabulary: The Isosceles Triangle Theorem states that the base angles of an isosceles triangle are congruent.

A more elaborate proof is then presented, involving a bisected line segment and perpendicular lines:

Given:

• BE bisects AD at C
• AB ⊥ BC
• DE ⊥ EC
• AB ≅ DE

Prove: ΔABC ≅ ΔDEC

This proof demonstrates how to combine multiple geometric concepts, including bisectors, perpendicular lines, and right angles, to set up the conditions for applying the HL Theorem.

Highlight: The HL Theorem can be used in conjunction with other geometric theorems and concepts to prove congruence in more complex scenarios.