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Congruent Triangles

A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

Congruent triangles have the same size and shape, which means corresponding sides and corresponding angles are equal. Triangle congruence shortcuts, such as SSS, SAS, ASA, and AAS, let you prove triangles are congruent without checking all six parts. This matters because once triangles are proven congruent, you can use corresponding parts to solve for missing measures. When practicing, match vertices in the correct order and identify which sides or angles correspond. A common mistake is assuming triangles are congruent from appearance alone instead of using a valid congruence reason.

What you'll learn

Why it matters: Manufacturing and construction use congruent triangles when matching braces, panels, or repeated parts that must fit the same way.

Worked example

Problem. Triangles ABC and DEF are congruent in that order. If AB = 9, what is DE?

  1. The order ABC matches DEF.
  2. Side AB corresponds to side DE.
  3. Corresponding sides are equal.

Answer: 9

Practice problems

1. Congruent triangles have...

Choices: Equal corresponding sides and angles · Only equal areas · Only equal perimeters · No matching parts

Show solution
  1. Warm-up: First identify exactly what the question is asking: Congruent triangles have...
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Congruence preserves side lengths and angle measures.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Equal corresponding sides and angles

2. Triangles ABC and DEF are congruent. If angle A is 40 degrees, what is angle D?

Show solution
  1. Warm-up: First identify exactly what the question is asking: Triangles ABC and DEF are congruent. If angle A is 40 degrees, what is angle D?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. A corresponds to D.
  4. Check the result by substituting or estimating: the response should match 40 and make sense in the original problem.

Answer: 40

3. Triangles ABC and DEF are congruent. If BC = 12, what is EF?

Show solution
  1. Core Practice: First identify exactly what the question is asking: Triangles ABC and DEF are congruent. If BC = 12, what is EF?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. BC corresponds to EF.
  4. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

4. If triangle CAT is congruent to triangle DOG, side CA corresponds to...

Choices: DO · OG · DG · GO

Show solution
  1. Core Practice: First identify exactly what the question is asking: If triangle CAT is congruent to triangle DOG, side CA corresponds to...
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. C matches D and A matches O.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: DO

5. Corresponding sides are 3x and 24. Find x.

Show solution
  1. Challenge: First identify exactly what the question is asking: Corresponding sides are 3x and 24. Find x.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 3x = 24.
  4. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

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