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Triangle Congruence Shortcuts

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

Triangle congruence is a claim about a specific correspondence between vertices. SSS, SAS, ASA, AAS, and the right-triangle hypotenuse-leg theorem provide sufficient conditions. SSA is not generally sufficient. A proof should list the established side or angle equalities, name the matching vertices in order, and only then use corresponding parts of congruent triangles.

What you'll learn

Why it matters: Bridge trusses and roof frames use triangle congruence evidence to verify repeated supports without measuring every part.

Worked example

Problem. In triangle ABC, AB = AC and D is the midpoint of BC. Prove that AD bisects angle BAC.

  1. AB = AC is given. BD = CD follows from D being the midpoint of BC. AD = AD is the reflexive property.
  2. Triangles ABD and ACD are congruent by SSS, with A corresponding to A, B to C, and D to D.
  3. Corresponding angles BAD and CAD are equal. This proves that AD bisects angle BAC.

Answer: Angle BAD = angle DAC, so AD bisects angle BAC.

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. Three pairs of congruent sides prove triangles congruent by...

Choices: SSS · SAS · ASA · AAA

Show solution
  1. SSS means side-side-side.

Answer: SSS

2. Two sides and the included angle prove triangles congruent by...

Choices: SAS · SSA · AAA · AAS

Show solution
  1. Included angle gives side-angle-side.

Answer: SAS

3. Two angles and the included side prove triangles congruent by...

Choices: ASA · SSS · SAS · AAA

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  1. Included side between two angles gives ASA.

Answer: ASA

4. Two angles and a non-included side prove triangles congruent by...

Choices: AAS · ASA · SSS · SSA

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  1. The side is not between the angles.

Answer: AAS

5. Which is not a valid general congruence shortcut?

Choices: SSA · SSS · SAS · ASA

Show solution
  1. SSA can be ambiguous.

Answer: SSA

Explain and apply

Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.

From congruence to perpendicularity

Draw triangle ABC with AB = AC and D the midpoint of BC. Use congruent triangles to prove AD is perpendicular to BC. State why you cannot simply assume from the sketch that angles ADB and ADC are right angles.

Review rubric and solution
  • The three side equalities AB = AC, BD = DC, and AD = AD are justified.
  • The congruence statement preserves the vertex correspondence.
  • The adjacent angles at D are proved equal, then shown to be supplementary.
  • Their 90-degree measure is deduced, not assumed from appearance.
  1. Triangles ABD and ACD are congruent by SSS, as in the worked example.
  2. Therefore angles ADB and ADC are equal by corresponding parts of congruent triangles.
  3. B, D, and C are collinear, so those angles sum to 180 degrees. Each must be 90 degrees, giving AD perpendicular to BC.
  4. The sketch is not proof of a right angle. The right angle is the conclusion established by congruence and the linear-pair relationship.

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