This 45-minute test covers conditional statements, related statements, counterexamples, inductive reasoning, properties of equality and proofs of the vertical angles theorem (G-CO.C.9). It has 8 questions worth 20 points. Justify every proof step.
- Grade: Geometry
- Chapter: Logic and Two-Column Proofs
- Common Core: G-CO.C.9
- Time: 45 minutes
- Total: 20 points (Question 1: 2 pts; Question 2: 3 pts; Question 3: 2 pts; Question 4: 2 pts; Question 5: 2 pts; Question 6: 3 pts; Question 7: 4 pts; Question 8: 2 pts)
Question 1: Parts of a conditional (2 points)
Consider: “If a quadrilateral has four right angles, then it is a rectangle.”
- Identify the hypothesis and the conclusion. (1 point)
- Write the converse and state whether it is true or false. (1 point)
Question 2: Related statements (3 points)
Consider: “If a figure is a rhombus, then it is a parallelogram.” For each part, write the statement and state whether it is true or false. Give a counterexample if it is false.
- The converse (1 point)
- The inverse (1 point)
- The contrapositive (1 point)
Question 3: Reasoning types and counterexample (2 points)
- Priya notices that the school bus was late on each of the last five Mondays and concludes it will be late next Monday. Is this inductive or deductive reasoning? (1 point)
- Find a counterexample to: “If \(x^{2}>x\), then \(x>1\).” (1 point)
Question 4: Inductive pattern (2 points)
The figure shows the first four patterns of dots.
- How many dots are in Pattern 5 and in Pattern 6? (1 point)
- Write a rule for the number of dots in Pattern n. (1 point)
Question 5: Properties of equality (2 points)
Name the property that justifies each statement.
- If \(7x=63\), then \(x=9\). (1 point)
- If \(AB=CD\) and \(CD=EF\), then \(AB=EF\). (1 point)
Question 6: Completing a proof (3 points)
Given: \(\angle A\) and \(\angle B\) are supplementary; \(m\angle A=72^{\circ}\). Prove: \(m\angle B=108^{\circ}\). Write the missing reason or statement for (a), (b) and (c).
| Statements | Reasons |
|---|---|
| 1. \(\angle A\) and \(\angle B\) are supplementary; \(m\angle A=72^{\circ}\) | 1. Given |
| 2. \(m\angle A+m\angle B=180^{\circ}\) | 2. (a) ____ |
| 3. (b) ____ | 3. Substitution Property of Equality |
| 4. \(m\angle B=108^{\circ}\) | 4. (c) ____ |
(1 point each)
Question 7: Vertical angles proof (4 points)
Lines AB and CD intersect at P. Write a two-column proof. Given: lines AB and CD intersect at P. Prove: \(\angle APD\) ≅ \(\angle BPC\).
Question 8: Vertical angles with algebra (2 points)
Two lines intersect. \(m\angle 1\)\(=(7x+3)^{\circ}\) and \(m\angle 3\)\(=(10x-18)^{\circ}\).
- Find x and name the theorem you used. (1 point)
- Find \(m\angle 2\) and give a reason. (1 point)
See every chapter on the Geometry math page.