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: $\triangle ABC \cong \triangle DEF$

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:

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?

💡 Show Solution

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)

2Problem 2easy

❓ Question:

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

💡 Show Solution

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

3Problem 3easy

❓ Question:

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

💡 Show Solution

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)

4Problem 4medium

❓ Question:

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

💡 Show Solution

We have:

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

The side $AB$ is included between the two angles $\angle A$ and $\angle B$ .

This is ASA (Angle-Side-Angle).

Answer: ASA

5Problem 5medium

❓ 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.

💡 Show Solution

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)

6Problem 6medium

❓ 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.

💡 Show Solution

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)

7Problem 7hard

❓ 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?

💡 Show Solution

Both are right triangles.

Given:

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

We have:

Congruent hypotenuses
Congruent legs

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

Answer: Yes, by HL theorem

8Problem 8hard

❓ Question:

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

💡 Show Solution

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

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