What is Congruent Triangles?

SSS, SAS, ASA, AAS, and HL theorems

How can I study Congruent Triangles effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 11 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

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What course covers Congruent Triangles?

Congruent Triangles is part of the Geometry course on Study Mondo, specifically in the Triangles section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Congruent Triangles?

Yes, this page includes 11 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Congruent Triangles

Definition

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

Symbol: $\triangle ABC \cong \triangle DEF$

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.

❓ Question:

Two triangles have all three pairs of corresponding sides equal: AB = DE = 5, BC = EF = 7, and AC = DF = 8. Are the triangles congruent? Which postulate proves it?

Step 1: Identify what is given: All three pairs of corresponding sides are equal: AB = DE = 5 BC = EF = 7 AC = DF = 8

Step 2: Recall SSS (Side-Side-Side) Congruence: If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent

Step 3: Apply SSS: Since all three pairs of sides are equal, △ABC ≅ △DEF by SSS

Step 4: What this means: The triangles are exactly the same size and shape All corresponding angles are also equal

Answer: Yes, the triangles are congruent by SSS (Side-Side-Side)

❓ Question:

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

We have three pairs of congruent sides:

$AB \cong DE$
$BC \cong EF$
$AC \cong DF$

This satisfies SSS (Side-Side-Side).

Answer: Yes, by SSS postulate

❓ Question:

In triangles ABC and XYZ: AB = XY = 10, AC = XZ = 12, and angle A = angle X = 60°. Are the triangles congruent? Which postulate?

Step 1: Identify what is given: Two sides and the included angle are equal: AB = XY = 10 (one side) Angle A = Angle X = 60° (included angle) AC = XZ = 12 (other side)

Step 2: Recall SAS (Side-Angle-Side) Congruence: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent

Step 3: Check that the angle is included: Angle A is between sides AB and AC ✓ Angle X is between sides XY and XZ ✓ The angle is included (between the two sides)

Step 4: Apply SAS: △ABC ≅ △XYZ by SAS

Answer: Yes, the triangles are congruent by SAS (Side-Angle-Side)

❓ Question:

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

We have three pairs of congruent sides:

$AB \cong DE$
$BC \cong EF$
$AC \cong DF$

❓ Question:

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

We have:

Two angles: $\angle A \cong \angle X$ and $\angle B \cong \angle Y$
One side: $AB \cong XY$

❓ Question:

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

We have:

Two angles: $\angle A \cong \angle X$ and $\angle B \cong \angle Y$
One side: $AB \cong XY$

❓ Question:

Triangles PQR and STU have: angle P = angle S = 45°, angle Q = angle T = 75°, and PQ = ST = 6. Prove the triangles are congruent.

Step 1: Identify what is given: Two angles and the included side: Angle P = Angle S = 45° Side PQ = Side ST = 6 (between angles P and Q) Angle Q = Angle T = 75°

Step 2: Recall ASA (Angle-Side-Angle) Congruence: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent

Step 3: Verify the side is included: PQ is the side between angles P and Q ✓ ST is the side between angles S and T ✓

Step 4: Apply ASA: △PQR ≅ △STU by ASA

Step 5: Additional note: We could also find the third angles: Angle R = 180° - 45° - 75° = 60° Angle U = 180° - 45° - 75° = 60°

Answer: The triangles are congruent by ASA (Angle-Side-Angle)

❓ Question:

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

Both are right triangles.

Given:

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

❓ Question:

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

Both are right triangles.

Given:

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

❓ Question:

In the figure, line segment AC bisects angle BAD and angle BCD. If AB = AD and CB = CD, prove that triangles ABC and ADC are congruent.

Step 1: List what we know: Given:

AC bisects angle BAD (so angle BAC = angle DAC)
AC bisects angle BCD (so angle BCA = angle DCA)
AB = AD
CB = CD
AC = AC (reflexive - shared side)

Step 2: Identify congruent parts: Sides:

AB = AD (given)
CB = CD (given)
AC = AC (reflexive property)

All three sides are congruent!

Step 3: Apply SSS Congruence: △ABC ≅ △ADC by SSS

Step 4: Alternative approach using SAS:

AB = AD (given)
Angle BAC = Angle DAC (AC bisects angle BAD)
AC = AC (reflexive) Therefore △ABC ≅ △ADC by SAS

Step 5: What this proves: BC = DC and all corresponding parts are congruent (CPCTC)

Answer: △ABC ≅ △ADC by SSS (or by SAS)

❓ Question:

Given: In quadrilateral ABCD, AB ∥ CD and AB = CD. The diagonals AC and BD intersect at point E. Prove that △ABE ≅ △CDE.

Step 1: Analyze the given information:

AB ∥ CD (parallel sides)
AB = CD (equal sides)
Need to prove △ABE ≅ △CDE

Step 2: Use properties of parallel lines: Since AB ∥ CD and AC is a transversal:

Angle BAE = Angle DCE (alternate interior angles)

Since AB ∥ CD and BD is a transversal:

Angle ABE = Angle CDE (alternate interior angles)

Step 3: Identify congruent parts: Angles:

Angle BAE = Angle DCE (alternate interior)
Angle ABE = Angle CDE (alternate interior) Side:
AB = CD (given)

Step 4: Apply ASA Congruence: We have:

Angle BAE = Angle DCE (angle)
AB = CD (side)
Angle ABE = Angle CDE (angle)

This is ASA: Angle-Side-Angle

Step 5: State the conclusion: △ABE ≅ △CDE by ASA

Step 6: Implications (CPCTC): Since the triangles are congruent:

AE = CE
BE = DE
The diagonals bisect each other

Answer: △ABE ≅ △CDE by ASA using alternate interior angles from parallel lines

Congruent Triangles

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Congruent Triangles

Definition

Congruence Postulates

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

📚 Practice Problems

1Problem 1easy

❓ Question:

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❓ Frequently Asked Questions

ASA (Angle-Side-Angle)

AAS (Angle-Angle-Side)

HL (Hypotenuse-Leg)

NOT Congruence Theorems

CPCTC

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4Problem 4easy

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5Problem 5medium

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6Problem 6medium

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8Problem 8hard

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10Problem 10medium

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