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Lesson 4.3: HL & CPCTC Diagram Exercises

Apply the Hypotenuse–Leg theorem for right triangles and use CPCTC to determine unknown sides and angles from congruent triangle diagrams.

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Lesson Description

This lesson introduces the Hypotenuse–Leg (HL) congruence theorem for right triangles and reinforces how CPCTC allows deduction of all corresponding sides and angles once triangle congruence is established.

Learning Objectives

  • Recognize right triangles suitable for HL congruence.
  • Identify hypotenuse and leg relationships in diagrams.
  • Apply CPCTC to determine unknown measures after congruence.

Topics Covered

  • Right triangle properties
  • Hypotenuse–Leg theorem (HL)
  • Diagram-based congruence reasoning
  • Using CPCTC to solve for unknown parts

Key Terms

  • Right Triangle: Triangle containing a 90° angle.
  • Hypotenuse: The side opposite the right angle.
  • Leg: Either side forming the right angle.
  • HL Theorem: Right triangles are congruent if the hypotenuse and one leg match.
  • CPCTC: Corresponding parts of congruent triangles are congruent.

Lesson Examples

Example 1: Using HL

  1. Confirm both triangles contain right angles.
  2. Check if the hypotenuse lengths are equal.
  3. Verify one corresponding leg is equal.
  4. Conclude triangles are congruent by HL.

Example 2: Applying CPCTC

  1. After proving triangle congruence, list matching vertices.
  2. Identify a corresponding angle or side not originally given.
  3. State that this part is congruent using CPCTC.

Practice Problems

  1. Given two right triangles with equal hypotenuse and one equal leg, prove they are congruent.
  2. After proving congruence, determine a missing side using CPCTC.
  3. Identify the hypotenuse in several diagram examples.
  4. Explain why HL applies only to right triangles.

Tips & Common Mistakes

  • Always verify the triangle is right before applying HL.
  • Do not confuse the hypotenuse with a regular side.
  • State congruence first, then apply CPCTC afterward.

Summary

The HL theorem provides a shortcut for proving right triangles congruent. Once congruence is proven, CPCTC allows determination of all remaining matching parts, making diagram problems much easier to solve.

Challenge Problems

  1. Create a diagram where HL proves triangle congruence but SAS cannot be applied directly.
  2. Write a full two-column proof using HL followed by CPCTC.
  3. Explain geometrically why knowing the hypotenuse and one leg fixes the entire right triangle.

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