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Grade 10 · unit review

Review: Geometry: Congruence and Bisector Reasoning

Congruence fixes a figure’s shape and size, while correspondence identifies matching vertices and sides. Perpendicular and angle bisectors encode equidistance or equal-angle properties that can be used and checked in a construction.

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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. Triangles have matching side lengths 7, 8, 9, and 7, 8, 9. Which criterion establishes congruence?
Need a hint?
  • Matching all three corresponding side lengths establishes triangle congruence through SSS.
  • A matching perimeter alone does not match individual sides.
2. Given triangle ABC congruent to triangle DEF in that order, AB=12, BC=13, AC=14. Find EF.
Need a hint?
  • The written order in triangle ABC congruent to DEF pairs A with D, B with E, and C with F.
  • Do not match sides by where they appear on a sketch.
3. Point P lies on the perpendicular bisector of segment AB. PA=14 cm. Find PB.
Need a hint?
  • A segment’s perpendicular bisector crosses its midpoint at a right angle.
  • Distance to either endpoint differs from distance to the midpoint.
4. A construction ray bisects an angle measuring 60°. What is either resulting angle?
Need a hint?
  • An angle bisector divides an angle into two congruent angles.
  • A ray inside an angle is not automatically a bisector.

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. Triangles have matching side lengths 9, 10, 11, and 9, 10, 11. Which criterion establishes congruence? Candidate answers: (A) AAA (B) Equal perimeter alone (C) SSA in every triangle (D) SSS. 2. Given triangle ABC congruent to triangle DEF in that order, AB=14, BC=15, AC=16. Find EF. Candidate answers: (A) 14 (B) 15 (C) 16 (D) 29. 3. Point P lies on the perpendicular bisector of segment AB. PA=18 cm. Find PB. Candidate answers: (A) 0 cm (B) 18 cm (C) 36 cm (D) 9 cm. 4. A construction ray bisects an angle measuring 68°. What is either resulting angle? Candidate answers: (A) 33 ° (B) 34 ° (C) 35 ° (D) 68 °.

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. SSS. Three corresponding side lengths match, which is SSS. Equal angles establish similarity, while equal perimeter alone does not determine shape. The supplied lengths satisfy triangle inequality because 9+10>11. 2. 15. The ordered correspondence is A↔D, B↔E, C↔F. Thus EF corresponds to BC, not AB or AC; EF=15. 3. 18 cm. Every point on a segment’s perpendicular bisector is equally distant from its two endpoints. Consequently PB=PA=18 cm. The theorem concerns endpoint distances, not distance to the midpoint. 4. 34 °. A bisector divides the angle into two congruent angles. Each is 68/2=34.0°. A line merely drawn inside an angle is not necessarily its bisector.

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