Congruent Triangles

SSS, SAS, ASA, AAS, and HL theorems

Congruent Triangles

Definition

Two triangles are congruent if all corresponding sides and angles are equal.

Symbol: △ABC≅△DEF\triangle ABC \cong \triangle DEF

Congruence Postulates

You don't need to show all 6 parts are equal. These shortcuts work:

SSS (Side-Side-Side)

If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

SAS (Side-Angle-Side)

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another, the triangles are congruent.

ASA (Angle-Side-Angle)

If two angles and the included side are congruent, the triangles are congruent.

AAS (Angle-Angle-Side)

If two angles and a non-included side are congruent, the triangles are congruent.

HL (Hypotenuse-Leg)

Right triangles only: If the hypotenuse and one leg are congruent, the triangles are congruent.

NOT Congruence Theorems

AAA - Shows similarity, not congruence SSA - Not sufficient (ambiguous case)

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

Once you prove triangles are congruent, you can conclude ALL corresponding parts are equal.

📚 Practice Problems

1Problem 1easy

❓ Question:

Can you prove △ABC≅△DEF\triangle ABC \cong \triangle DEF if AB=DE=5AB = DE = 5, BC=EF=7BC = EF = 7, and AC=DF=6AC = DF = 6?

💡 Show Solution

We have three pairs of congruent sides:

  • AB≅DEAB \cong DE
  • BC≅EFBC \cong EF
  • AC≅DFAC \cong DF

This satisfies SSS (Side-Side-Side).

Answer: Yes, by SSS postulate

2Problem 2medium

❓ Question:

Given: AB≅XYAB \cong XY, ∠A≅∠X\angle A \cong \angle X, ∠B≅∠Y\angle B \cong \angle Y. Which congruence postulate proves △ABC≅△XYZ\triangle ABC \cong \triangle XYZ?

💡 Show Solution

We have:

  • Two angles: ∠A≅∠X\angle A \cong \angle X and ∠B≅∠Y\angle B \cong \angle Y
  • One side: AB≅XYAB \cong XY

The side ABAB is included between the two angles ∠A\angle A and ∠B\angle B.

This is ASA (Angle-Side-Angle).

Answer: ASA

3Problem 3hard

❓ Question:

In right triangles △PQR\triangle PQR and △STU\triangle STU (right angles at QQ and TT), PR=SU=10PR = SU = 10 and QR=TU=6QR = TU = 6. Are the triangles congruent? If so, by what theorem?

💡 Show Solution

Both are right triangles.

Given:

  • PR=SU=10PR = SU = 10 (these are the hypotenuses)
  • QR=TU=6QR = TU = 6 (these are legs)

We have:

  • Congruent hypotenuses
  • Congruent legs

This satisfies HL (Hypotenuse-Leg) for right triangles.

Answer: Yes, by HL theorem

🎴

Practice with Flashcards

Review key concepts with our flashcard system

📖

Browse All Topics

Explore other calculus topics