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Grade 10 · unit review

Review: Geometry: Angle Theorems and Triangle Constraints

Parallel-line and triangle theorems connect geometric conditions to numerical equations. Some side and angle values are impossible, so reasoning must check the defining configuration rather than rely only on arithmetic.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Two parallel lines are cut by a transversal. One angle is 45°. Find its corresponding angle.
Need a hint?
  • A transversal crossing parallel lines creates equal corresponding angles.
  • Locate corresponding positions at the two crossings before equating angles.
2. A triangle has interior angles 45° and 65°. Find the third.
Need a hint?
  • A Euclidean triangle’s interior angles sum to 180 degrees.
  • Use interior angles only, not a nearby exterior angle.
3. An isosceles triangle has vertex angle 50°. Find either equal base angle.
Need a hint?
  • Equal sides in an isosceles triangle face equal base angles.
  • The vertex angle lies between the equal sides.
4. Two triangle sides have lengths 7 and 8. Which proposed third length makes a nondegenerate triangle?
Need a hint?
  • A nondegenerate triangle’s third side must be less than the sum and greater than the absolute difference of the other two sides.
  • Equality creates a collapsed straight arrangement rather than a triangle with area.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Two parallel lines are cut by a transversal. One angle is 49°. Find its corresponding angle. Candidate answers: (A) 131 ° (B) 41 ° (C) 49 ° (D) 59 °. 2. A triangle has interior angles 47° and 67°. Find the third. Candidate answers: (A) 132 ° (B) 65 ° (C) 66 ° (D) 67 °. 3. An isosceles triangle has vertex angle 54°. Find either equal base angle. Candidate answers: (A) 126 ° (B) 62 ° (C) 63 ° (D) 64 °. 4. Two triangle sides have lengths 9 and 10. Which proposed third length makes a nondegenerate triangle? Candidate answers: (A) 1 (B) 18 (C) 19 (D) 20.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 49 °. Corresponding angles are equal when the lines are parallel. Therefore the corresponding angle is 49°. Without parallelism, this inference would not be warranted. 2. 66 °. The three interior angles of a Euclidean triangle total 180°. Subtract both known values to obtain 66°. The answer is positive for this supplied triangle. 3. 63 °. The two base angles are equal. They total 180−54=126°, so each is half, 63.0°. The vertex angle is the angle between the equal sides. 4. 18. The third side z must satisfy |10−9|<z<9+10, or 1<z<19. The value 18 is inside; equality would make a collapsed straight arrangement.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.