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Triangle Theorems Practice Tests

These Grade 10 Geometry practice tests develop theorem-based reasoning with triangle angles, side lengths, centers, inequalities, right-triangle formulas, and proof applications.

Skills covered

  • Applying the Triangle Sum, Exterior Angle, Isosceles, Larger Angle-Larger Side, and Triangle Inequality Theorems
  • Identifying medians, altitudes, angle bisectors, perpendicular bisectors, centroid, circumcenter, incenter, and orthocenter
  • Using the Pythagorean Theorem, its converse, geometric means, and 30-60-90 and 45-45-90 relationships
  • Solving side-length, angle-measure, perimeter, and exterior-angle problems
  • Using triangle theorems in congruence proofs, coordinate geometry, construction, and measurement applications

How to use these practice tests

Underline the givens and identify the theorem before calculating. For side-length questions, check the triangle inequality; for right triangles, label the hypotenuse before applying the formula.

For more Grade 10 Math practice, return to the Grade 10 Math category.

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Grade 10 Triangle Theorems Practice Tests

Practice Grade 10 Geometry questions about triangle angle and side theorems, centers, inequalities, right triangles, and proof applications.

Choose the best answer. Identify the relevant triangle theorem, substitute known measures, and check the result using a diagram, inequality, or algebraic relationship.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

What is the Pythagorean Theorem?

Please select an answer first.

Correct answer: A

The correct answer is a2+b2=c2a^2+b^2=c^2 for a right triangle..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: For a right triangle, the squares of the leg lengths add to the square of the hypotenuse.
  3. Check the conclusion: The correct answer is a2+b2=c2a^2+b^2=c^2 for a right triangle..

For a right triangle, the squares of the leg lengths add to the square of the hypotenuse.

2. QuestionMultiple choice

Which side is the hypotenuse of a right triangle?

Please select an answer first.

Correct answer: B

The correct answer is The side opposite the right angle..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: The hypotenuse is always opposite the 90-degree angle and is the longest side.
  3. Check the conclusion: The correct answer is The side opposite the right angle..

The hypotenuse is always opposite the 90-degree angle and is the longest side.

3. QuestionMultiple choice

A right triangle has legs 6 and 8. What is its hypotenuse?

Please select an answer first.

Correct answer: C

The correct answer is 10..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: Use c=62+82=100=10c=\sqrt{6^2+8^2}=\sqrt{100}=10.
  3. Check the conclusion: The correct answer is 10..

Use c=62+82=100=10c=\sqrt{6^2+8^2}=\sqrt{100}=10.

4. QuestionMultiple choice

A right triangle has hypotenuse 13 and one leg 5. What is the other leg?

Please select an answer first.

Correct answer: D

The correct answer is 12..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: Use x=132−52=144=12x=\sqrt{13^2-5^2}=\sqrt{144}=12.
  3. Check the conclusion: The correct answer is 12..

Use x=132−52=144=12x=\sqrt{13^2-5^2}=\sqrt{144}=12.

5. QuestionMultiple choice

What does the converse of the Pythagorean Theorem show?

Please select an answer first.

Correct answer: A

The correct answer is If the square of the longest side equals the sum of the other two squares, the triangle is right..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: The converse identifies a right triangle when c2=a2+b2c^2=a^2+b^2 holds for the longest side.
  3. Check the conclusion: The correct answer is If the square of the longest side equals the sum of the other two squares, the triangle is right..

The converse identifies a right triangle when c2=a2+b2c^2=a^2+b^2 holds for the longest side.

6. QuestionMultiple choice

In a 30-60-90 triangle with hypotenuse 14, what is the short leg?

Please select an answer first.

Correct answer: B

The correct answer is 7..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: In a 30-60-90 triangle, the short leg is half the hypotenuse: 14 ÷ 2 = 7.
  3. Check the conclusion: The correct answer is 7..

In a 30-60-90 triangle, the short leg is half the hypotenuse: 14 ÷ 2 = 7.

7. QuestionMultiple choice

In a 45-45-90 triangle with leg 5, what is the hypotenuse?

Please select an answer first.

Correct answer: C

The correct answer is 525\sqrt{2}..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: In a 45-45-90 triangle, the hypotenuse equals a leg times 2\sqrt{2}.
  3. Check the conclusion: The correct answer is 525\sqrt{2}..

In a 45-45-90 triangle, the hypotenuse equals a leg times 2\sqrt{2}.

8. QuestionMultiple choice

A right triangle altitude creates segments 4 and 9 on the hypotenuse. What is the altitude length?

Please select an answer first.

Correct answer: D

The correct answer is 6..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: The geometric-mean relationship is h2=pqh^2=pq, so h=4⋅9=6h=\sqrt{4\cdot9}=6.
  3. Check the conclusion: The correct answer is 6..

The geometric-mean relationship is h2=pqh^2=pq, so h=4⋅9=6h=\sqrt{4\cdot9}=6.

9. QuestionMultiple choice

Is a triangle with side lengths 7, 24, and 25 a right triangle?

Please select an answer first.

Correct answer: A

The correct answer is Yes, because 72+242=2527^2+24^2=25^2..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: Since 49 + 576 = 625, the converse of the Pythagorean Theorem proves it is right.
  3. Check the conclusion: The correct answer is Yes, because 72+242=2527^2+24^2=25^2..

Since 49 + 576 = 625, the converse of the Pythagorean Theorem proves it is right.

10. QuestionMultiple choice

Which set of lengths does not form a right triangle?

Please select an answer first.

Correct answer: B

The correct answer is 5, 6, and 8..

  1. Identify the triangle information: List the angle measures, side lengths, center, segment, or proof relationship given in the problem.
  2. Apply the triangle theorem: For 5, 6, 8, 5² + 6² = 61, not 8² = 64, so it is not a right triangle.
  3. Check the conclusion: The correct answer is 5, 6, and 8..

For 5, 6, 8, 5² + 6² = 61, not 8² = 64, so it is not a right triangle.

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