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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 a median of a triangle?

Please select an answer first.

Correct answer: A

The correct answer is A segment from a vertex to the midpoint of the opposite side..

  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: A median joins a vertex to the midpoint of the opposite side.
  3. Check the conclusion: The correct answer is A segment from a vertex to the midpoint of the opposite side..

A median joins a vertex to the midpoint of the opposite side.

2. QuestionMultiple choice

What is an altitude of a triangle?

Please select an answer first.

Correct answer: B

The correct answer is A perpendicular segment from a vertex to the opposite side or its extension..

  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: An altitude is drawn perpendicular to the line containing the opposite side.
  3. Check the conclusion: The correct answer is A perpendicular segment from a vertex to the opposite side or its extension..

An altitude is drawn perpendicular to the line containing the opposite side.

3. QuestionMultiple choice

What is an angle bisector in a triangle?

Please select an answer first.

Correct answer: C

The correct answer is A segment that divides a vertex angle into two congruent angles..

  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: A triangle angle bisector splits one interior angle into two equal measures.
  3. Check the conclusion: The correct answer is A segment that divides a vertex angle into two congruent angles..

A triangle angle bisector splits one interior angle into two equal measures.

4. QuestionMultiple choice

What is a perpendicular bisector of a side?

Please select an answer first.

Correct answer: D

The correct answer is A line perpendicular to the side at its midpoint..

  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: A perpendicular bisector meets a segment at a right angle exactly at its midpoint.
  3. Check the conclusion: The correct answer is A line perpendicular to the side at its midpoint..

A perpendicular bisector meets a segment at a right angle exactly at its midpoint.

5. QuestionMultiple choice

How does the centroid divide each median?

Please select an answer first.

Correct answer: A

The correct answer is It divides the median in a 2:1 ratio, with the longer part next to the vertex..

  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 centroid is two-thirds of the way from a vertex along a median, giving a 2:1 division.
  3. Check the conclusion: The correct answer is It divides the median in a 2:1 ratio, with the longer part next to the vertex..

The centroid is two-thirds of the way from a vertex along a median, giving a 2:1 division.

6. QuestionMultiple choice

What is the circumcenter of a triangle?

Please select an answer first.

Correct answer: B

The correct answer is The intersection of the perpendicular bisectors of its sides..

  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 perpendicular bisectors meet at the circumcenter, which is equidistant from the three vertices.
  3. Check the conclusion: The correct answer is The intersection of the perpendicular bisectors of its sides..

The perpendicular bisectors meet at the circumcenter, which is equidistant from the three vertices.

7. QuestionMultiple choice

What is the incenter of a triangle?

Please select an answer first.

Correct answer: C

The correct answer is The intersection of the three angle bisectors..

  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 angle bisectors meet at the incenter, which is equidistant from the sides.
  3. Check the conclusion: The correct answer is The intersection of the three angle bisectors..

The angle bisectors meet at the incenter, which is equidistant from the sides.

8. QuestionMultiple choice

What is the orthocenter of a triangle?

Please select an answer first.

Correct answer: D

The correct answer is The intersection of the three altitudes or their extensions..

  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 altitudes of a triangle meet at the orthocenter, inside or outside depending on the triangle.
  3. Check the conclusion: The correct answer is The intersection of the three altitudes or their extensions..

The altitudes of a triangle meet at the orthocenter, inside or outside depending on the triangle.

9. QuestionMultiple choice

A triangle midsegment has length 9. What is the length of the parallel third side?

Please select an answer first.

Correct answer: A

The correct answer is 18..

  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: A midsegment is half the length of the third side, so the third side is 2 · 9 = 18.
  3. Check the conclusion: The correct answer is 18..

A midsegment is half the length of the third side, so the third side is 2 · 9 = 18.

10. QuestionMultiple choice

A point equidistant from all three sides of a triangle is the...

Please select an answer first.

Correct answer: B

The correct answer is Incenter..

  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 incenter lies on all three angle bisectors and has equal perpendicular distances to the sides.
  3. Check the conclusion: The correct answer is Incenter..

The incenter lies on all three angle bisectors and has equal perpendicular distances to the sides.

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