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Quiz: Triangle Congruence and Properties

Test your understanding of triangle classifications, congruence criteria, and triangle theorems with these questions.

1. A triangle with side lengths 5 cm, 5 cm, and 7 cm is classified as:

  1. Scalene
  2. Isosceles
  3. Equilateral
  4. Right
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The correct answer is B. An isosceles triangle has exactly two sides of equal length. This triangle has two sides measuring 5 cm (congruent) and one side measuring 7 cm, making it isosceles. A scalene triangle (A) has all sides different. An equilateral triangle (C) has all three sides equal. "Right" (D) is a classification by angles, not sides.

Concept Tested: Triangle Classification by Sides

See: Chapter 7: Classifying Triangles by Sides

2. What does SSS stand for in triangle congruence?

  1. Same Side Segments
  2. Side-Side-Side
  3. Similar Shape Standard
  4. Supplementary Side Sets
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The correct answer is B. SSS stands for Side-Side-Side, one of the five triangle congruence criteria. The SSS Congruence Postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. This is one of the fundamental ways to prove triangles are identical in size and shape.

Concept Tested: SSS Congruence

See: Chapter 7: SSS Congruence

3. The sum of the three interior angles of any triangle is always:

  1. 90°
  2. 180°
  3. 270°
  4. 360°
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The correct answer is B. The Triangle Sum Theorem states that the sum of the three interior angles of any triangle always equals 180°. This is a fundamental property of triangles in Euclidean geometry and holds true for all triangles—acute, right, obtuse, scalene, isosceles, and equilateral.

Concept Tested: Triangle Sum Theorem

See: Chapter 7: Triangle Sum Theorem

4. Why is AAA (Angle-Angle-Angle) NOT sufficient to prove triangle congruence?

  1. Because all triangles have the same angles
  2. Because triangles with equal angles can have different sizes
  3. Because angles don't matter for congruence
  4. Because you need at least four pieces of information
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The correct answer is B. AAA proves that triangles are similar (same shape) but not necessarily congruent (same size and shape). Two triangles can have all three pairs of angles equal but have different side lengths—one could be a scaled version of the other. For congruence, we need information about sides as well. Option A is false—triangles have different angle combinations. Option C is wrong—angles do matter, but aren't sufficient alone. Option D is incorrect—some valid criteria use only three pieces.

Concept Tested: Invalid Congruence Criteria (AAA)

See: Chapter 7: Triangle Congruence Criteria

5. In the SAS (Side-Angle-Side) congruence criterion, what does "included angle" mean?

  1. The largest angle in the triangle
  2. Any angle in the triangle
  3. The angle between the two given sides
  4. The angle opposite the longest side
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The correct answer is C. The included angle is the angle that is between (formed by) the two given sides. For SAS to work, the angle must be included—sandwiched between the two sides you're comparing. If you have two sides and a non-included angle, that's SSA, which is NOT sufficient for proving congruence. The included angle is the key to making SAS valid.

Concept Tested: SAS Congruence (Included Angle)

See: Chapter 7: SAS Congruence

6. What is the difference between an isosceles triangle and an equilateral triangle?

  1. Isosceles has two equal sides; equilateral has all three sides equal
  2. Isosceles has all equal angles; equilateral has different angles
  3. Isosceles is always acute; equilateral can be obtuse
  4. There is no difference; they are the same
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The correct answer is A. An isosceles triangle has exactly two sides of equal length, while an equilateral triangle has all three sides equal. Because an equilateral triangle has all sides equal, it's a special case of isosceles—every equilateral triangle is also isosceles, but not every isosceles triangle is equilateral. Option B is backwards. Option C is false—equilateral triangles are always acute (all 60° angles). Option D is incorrect.

Concept Tested: Isosceles vs Equilateral Triangles

See: Chapter 7: Triangle Classification

7. A triangle has angles measuring 45°, 55°, and 80°. This triangle is classified as:

  1. Right triangle
  2. Obtuse triangle
  3. Acute triangle
  4. Equilateral triangle
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The correct answer is C. An acute triangle has all three angles less than 90°. Since 45°, 55°, and 80° are all less than 90°, this is an acute triangle. A right triangle (A) has one 90° angle. An obtuse triangle (B) has one angle greater than 90°. An equilateral triangle (D) is a classification by sides, not angles (though all equilateral triangles are acute).

Concept Tested: Triangle Classification by Angles

See: Chapter 7: Classifying Triangles by Angles

8. Two triangles have the following congruent parts: AB ≅ DE, ∠A ≅ ∠D, and ∠B ≅ ∠E. Which congruence criterion proves the triangles are congruent?

  1. SSS
  2. SAS
  3. ASA
  4. HL
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The correct answer is C. We have two angles (∠A ≅ ∠D and ∠B ≅ ∠E) and the included side between them (AB ≅ DE). This matches the ASA (Angle-Side-Angle) criterion—two angles and the included side. SSS (A) requires three sides. SAS (B) requires two sides and the included angle. HL (D) is only for right triangles with hypotenuse and leg.

Concept Tested: ASA Congruence

See: Chapter 7: ASA Congruence

9. In a triangle, two angles measure 65° and 48°. What is the measure of the third angle?

  1. 23°
  2. 67°
  3. 113°
  4. 167°
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The correct answer is B. Using the Triangle Sum Theorem, the three angles must add to 180°. If two angles are 65° and 48°, then: 65° + 48° + x = 180°, so 113° + x = 180°, giving x = 67°. Option A (23°) results from incorrect subtraction. Option C (113°) is the sum of the two given angles, not the third angle. Option D (167°) results from adding instead of finding the difference.

Concept Tested: Triangle Sum Theorem (Application)

See: Chapter 7: Triangle Sum Theorem

10. Why is SSA (Side-Side-Angle) NOT a valid triangle congruence criterion?

  1. Because you always need at least three angles
  2. Because side lengths don't matter for congruence
  3. Because two different triangles can have the same SSA measurements
  4. Because all triangles with SSA are similar, not congruent
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The correct answer is C. SSA is called the "ambiguous case" because given two sides and a non-included angle, it's possible to construct two different triangles that satisfy those conditions. Since the same measurements can produce different triangles, SSA doesn't guarantee congruence. Option A is false—SSS uses no angles. Option B is wrong—sides do matter. Option D confuses SSA with AAA (which gives similarity).

Concept Tested: Invalid Congruence Criteria (SSA)

See: Chapter 7: Triangle Congruence Criteria

Score

Check your answers above. How did you do?

  • 9-10 correct: Excellent! You have mastered triangle classifications and congruence criteria.
  • 7-8 correct: Good work! Review the five congruence criteria and their requirements.
  • 5-6 correct: Study triangle classifications and practice identifying congruence criteria.
  • Below 5: Review the chapter carefully, especially SSS, SAS, ASA, AAS, HL and why AAA and SSA don't work.