Coordinate Proofs with Slope
A free Geometry lesson from the “Coordinate Geometry and Proof” unit, with a worked example and practice problems including step-by-step solutions.
Coordinate proofs use algebraic evidence to prove geometric facts. Slopes show whether sides are parallel or perpendicular, and distances show whether sides are congruent. In Coordinate Geometry and Proof, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.
What you'll learn
- Use slope for parallel sides
- Use slope for perpendicular sides
- Classify coordinate figures
Worked example
Problem. A quadrilateral has opposite sides with slopes 2 and 2. What does that show?
- Lines with equal slopes are parallel if they are distinct.
- The opposite sides have the same slope.
- So those sides are parallel.
- Connect the result back to Coordinate Proofs with Slope so the geometric relationship is explicit.
Answer: Those opposite sides are parallel.
Practice problems
1. Lines with slopes 3 and 3 are...
Choices: Parallel · Perpendicular · Neither · Vertical
Show solution
- Warm-up: First identify exactly what the question is asking: Lines with slopes 3 and 3 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 are parallel.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: Parallel
2. Lines with slopes 2 and -1/2 are...
Choices: Perpendicular · Parallel · Neither · Horizontal
Show solution
- Warm-up: First identify exactly what the question is asking: Lines with slopes 2 and -1/2 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.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: Perpendicular
3. What slope is perpendicular to 4?
Show solution
- Core Practice: First identify exactly what the question is asking: What slope is perpendicular to 4?
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Use the opposite reciprocal.
- Check the result by substituting or estimating: the response should match -1/4 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: -1/4
4. A quadrilateral has both pairs of opposite sides parallel. It is at least a...
Choices: Parallelogram · Triangle · Circle · Kite only
Show solution
- Core Practice: First identify exactly what the question is asking: A quadrilateral has both pairs of opposite sides parallel. It is at least a...
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- This is the definition of a parallelogram.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: Parallelogram
5. A quadrilateral has four congruent sides. It is a...
Choices: Rhombus · Rectangle always · Circle · Triangle
Show solution
- Challenge: First identify exactly what the question is asking: A quadrilateral has four congruent sides. It is a...
- For congruence, match corresponding sides and angles in the same order before choosing a theorem or equation.
- A rhombus has four congruent sides.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: Rhombus
Practice this interactively with instant feedback and an AI tutor.
Practice Coordinate Proofs with Slope Take the free placement check