This geometry reasoning proof test answer key gives the full solution and the scoring guide for each of the 8 test questions on logic and two-column proofs.
Each answer gives a model solution and a scoring guide. Award credit for correct reasoning even when the wording differs. In proofs, every statement needs a valid reason, and steps may appear in any logical order that works.
Geometry Reasoning Proof Test Answer Key: questions 1–5
Question 1: Parts of a conditional (2 points)
- Hypothesis: a quadrilateral has four right angles. Conclusion: it is a rectangle.
- Converse: If a quadrilateral is a rectangle, then it has four right angles. This is true by the definition of a rectangle.
Scoring: 1 point for both parts identified; 1 point for the correct converse with the correct truth value.
Question 2: Related statements (3 points)
- Converse: If a figure is a parallelogram, then it is a rhombus. False. A rectangle that is 5 cm by 3 cm is a parallelogram but not a rhombus.
- Inverse: If a figure is not a rhombus, then it is not a parallelogram. False. The same 5 cm by 3 cm rectangle is not a rhombus but is a parallelogram.
- Contrapositive: If a figure is not a parallelogram, then it is not a rhombus. True, because the original statement is true.
Scoring: 1 point per part for the correct statement, correct truth value and, for false statements, a valid counterexample.
Question 3: Reasoning types and counterexample (2 points)
- Inductive: the conclusion comes from a pattern of observed cases.
- Let \(x=-2\). Then \(x^{2}=4>-2\), so the hypothesis is true, but \(-2>1\) is false. \(x=-2\) is a counterexample (other negative numbers and numbers between 0 and 1 that fit the hypothesis also work).
Scoring: 1 point for each part. In part (b), the value must make the hypothesis true and the conclusion false.
Question 4: Inductive pattern (2 points)
- The counts are 3, 8, 15, 24, so the differences are 5, 7, 9 and the next are 11 and 13: \(24+11=35\) and \(35+13=48\). 35 dots and 48 dots.
- Pattern n has n rows of n + 2 dots: \(n(n+2)\). Check: \(5\times 7=35\).
Scoring: 1 point for both counts; 1 point for a correct rule written with n.
Question 5: Properties of equality (2 points)
- Division Property of Equality.
- Transitive Property of Equality.
Scoring: 1 point for each correct property.
Geometry Reasoning Proof Test Answer Key: questions 6–8
Question 6: Completing a proof (3 points)
- Definition of supplementary angles.
- \(72^{\circ}+m\angle B=180^{\circ}\).
- Subtraction Property of Equality (subtract 72 from both sides).
Check: \(72+108=180\).
Scoring: 1 point for each correct entry.
Question 7: Vertical angles proof (4 points)
| Statements | Reasons |
|---|---|
| 1. Lines AB and CD intersect at P | 1. Given |
| 2. \(\angle APD\) and \(\angle DPB\) are a linear pair; \(\angle DPB\) and \(\angle BPC\) are a linear pair | 2. Definition of linear pair |
| 3. \(\angle APD\) and \(\angle DPB\) are supplementary; \(\angle DPB\) and \(\angle BPC\) are supplementary | 3. Linear Pair Postulate |
| 4. \(m\angle APD+m\angle DPB=180^{\circ}\); \(m\angle DPB+m\angle BPC=180^{\circ}\) | 4. Definition of supplementary angles |
| 5. \(m\angle APD+m\angle DPB=m\angle DPB+m\angle BPC\) | 5. Transitive Property of Equality |
| 6. \(m\angle APD=m\angle BPC\) | 6. Subtraction Property of Equality |
| 7. \(\angle APD\) ≅ \(\angle BPC\) | 7. Definition of congruent angles |
Scoring: 1 point for naming the linear pairs with a reason (steps 2-3); 1 point for the two sums equal to 180 (step 4); 1 point for combining them and subtracting (steps 5-6); 1 point for the conclusion with a reason (step 7). Accept an equivalent proof with every statement justified.
Question 8: Vertical angles with algebra (2 points)
- Vertical angles are congruent (Vertical Angles Theorem): \(7x+3=10x-18\), so \(21=3x\) and x = 7.
- \(m\angle 1\)\(=7(7)+3=52^{\circ}\). Angles 1 and 2 are a linear pair, so they are supplementary: \(m\angle 2\)\(=180-52=128^{\circ}\). 128°.
Scoring: 1 point for x = 7 with the theorem named; 1 point for 128° with the linear pair reason.
Scoring summary
20 points: add the points of every question of this geometry reasoning proof test answer key.
How to use this geometry reasoning proof test answer key
Use this geometry reasoning proof test answer key to score the test fairly and to see which skills need more practice.
See every chapter on the Geometry math page.