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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

How can two congruent base angles prove a triangle is isosceles?

Please select an answer first.

Correct answer: A

The correct answer is The converse of the Isosceles Triangle Theorem makes the opposite sides congruent..

  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: Congruent angles have congruent opposite sides by the converse of the Isosceles Triangle Theorem.
  3. Check the conclusion: The correct answer is The converse of the Isosceles Triangle Theorem makes the opposite sides congruent..

Congruent angles have congruent opposite sides by the converse of the Isosceles Triangle Theorem.

2. QuestionMultiple choice

If D is the midpoint of BC and AB ≅ AC, which criterion can prove △ABD ≅ △ACD?

Please select an answer first.

Correct answer: B

The correct answer is SSS..

  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: AB ≅ AC, BD ≅ DC because D is a midpoint, and AD is shared; therefore SSS applies.
  3. Check the conclusion: The correct answer is SSS..

AB ≅ AC, BD ≅ DC because D is a midpoint, and AD is shared; therefore SSS applies.

3. QuestionMultiple choice

If point P lies on the perpendicular bisector of AB, what is true?

Please select an answer first.

Correct answer: C

The correct answer is PA ≅ PB..

  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 Bisector Theorem gives equal distances from P to A and B.
  3. Check the conclusion: The correct answer is PA ≅ PB..

The Perpendicular Bisector Theorem gives equal distances from P to A and B.

4. QuestionMultiple choice

If point P lies on an angle bisector, what can be true about its distances to the two sides?

Please select an answer first.

Correct answer: D

The correct answer is Its perpendicular distances to the two sides are equal..

  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: Points on an angle bisector are equidistant from the two sides of the angle.
  3. Check the conclusion: The correct answer is Its perpendicular distances to the two sides are equal..

Points on an angle bisector are equidistant from the two sides of the angle.

5. QuestionMultiple choice

A triangle midsegment is parallel to a base of length 36. What is the midsegment length?

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 parallel third side: 36 ÷ 2 = 18.
  3. Check the conclusion: The correct answer is 18..

A midsegment is half the length of the parallel third side: 36 ÷ 2 = 18.

6. QuestionMultiple choice

A 6-foot ladder reaches a point 8 feet high on a wall. How long is the ladder?

Please select an answer first.

Correct answer: B

The correct answer is 10 feet..

  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 ladder is the hypotenuse of a 6-8 right triangle, so its length is 10 feet.
  3. Check the conclusion: The correct answer is 10 feet..

The ladder is the hypotenuse of a 6-8 right triangle, so its length is 10 feet.

7. QuestionMultiple choice

The points (0,0), (4,0), and (4,3) form which type of triangle?

Please select an answer first.

Correct answer: C

The correct answer is A right scalene 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: The horizontal and vertical legs meet at a right angle, and the side lengths 3, 4, and 5 are all different.
  3. Check the conclusion: The correct answer is A right scalene triangle..

The horizontal and vertical legs meet at a right angle, and the side lengths 3, 4, and 5 are all different.

8. QuestionMultiple choice

Why is a triangular frame rigid when its side lengths are fixed?

Please select an answer first.

Correct answer: D

The correct answer is SSS determines a unique triangle up to congruence..

  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: SSS congruence means fixed three side lengths determine the triangle’s shape.
  3. Check the conclusion: The correct answer is SSS determines a unique triangle up to congruence..

SSS congruence means fixed three side lengths determine the triangle’s shape.

9. QuestionMultiple choice

A cable forms the hypotenuse of a right triangle with legs 5 m and 12 m. How long is the cable?

Please select an answer first.

Correct answer: A

The correct answer is 13 m..

  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 Pythagorean Theorem gives 5-12-13, so the cable is 13 meters.
  3. Check the conclusion: The correct answer is 13 m..

The Pythagorean Theorem gives 5-12-13, so the cable is 13 meters.

10. QuestionMultiple choice

Two right triangles have congruent hypotenuses and one congruent leg. Which theorem proves them congruent?

Please select an answer first.

Correct answer: B

The correct answer is HL..

  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-Leg theorem applies specifically to right triangles with a congruent hypotenuse and leg.
  3. Check the conclusion: The correct answer is HL..

The Hypotenuse-Leg theorem applies specifically to right triangles with a congruent hypotenuse and leg.

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