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.
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.
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.
20 points
See every chapter on the Geometry math page.