Coordinate Quadrilateral Proofs
A free Geometry lesson from the “Coordinate Geometry and Proof” unit, with a worked example and practice problems including step-by-step solutions.
Coordinate quadrilateral proofs combine slope and distance. Equal slopes prove parallel sides, opposite reciprocal slopes prove perpendicular sides, and equal distances prove congruent sides.
What you'll learn
- Use slope to prove parallel and perpendicular sides
- Use distance to prove congruent sides
- Classify quadrilaterals on the coordinate plane
Worked example
Problem. A quadrilateral has both pairs of opposite sides with equal slopes. What can you prove?
- Equal slopes show the opposite sides are parallel.
- Both pairs of opposite sides are parallel.
- A quadrilateral with both pairs of opposite sides parallel is a parallelogram.
Answer: It is a parallelogram.
Practice problems
1. Equal slopes prove lines are...
Choices: Parallel · Perpendicular · Congruent · Supplementary
Show solution
- Warm-up: First identify exactly what the question is asking: Equal slopes prove lines are...
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Equal slopes mean parallel lines if distinct.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Parallel
2. Slopes 3 and -1/3 prove lines are...
Choices: Perpendicular · Parallel · Neither · Horizontal
Show solution
- Warm-up: First identify exactly what the question is asking: Slopes 3 and -1/3 prove lines are...
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- They are opposite reciprocals.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Perpendicular
3. Both pairs of opposite sides parallel proves a quadrilateral is a...
Choices: Parallelogram · Circle · Triangle · Ray
Show solution
- Core Practice: First identify exactly what the question is asking: Both pairs of opposite sides parallel proves a quadrilateral is a...
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- This is the parallelogram definition.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Parallelogram
4. Four congruent sides prove a quadrilateral is at least a...
Choices: Rhombus · Rectangle · Trapezoid · Line
Show solution
- Core Practice: First identify exactly what the question is asking: Four congruent sides prove a quadrilateral is at least a...
- For congruence, match corresponding sides and angles in the same order before choosing a theorem or equation.
- A rhombus has all sides congruent.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Rhombus
5. A parallelogram with perpendicular adjacent sides is a...
Choices: Rectangle · Rhombus always · Trapezoid · Pentagon
Show solution
- Challenge: First identify exactly what the question is asking: A parallelogram with perpendicular adjacent sides is a...
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Perpendicular adjacent sides create right angles.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Rectangle
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