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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

Why it matters: Civil engineers and map designers use slopes to show when roads, rails, or edges are parallel or perpendicular.

Worked example

Problem. A quadrilateral has opposite sides with slopes 2 and 2. What does that show?

  1. Lines with equal slopes are parallel if they are distinct.
  2. The opposite sides have the same slope.
  3. So those sides are parallel.
  4. Connect the result back to Coordinate Proofs with Slope so the geometric relationship is explicit.

Answer: Those opposite sides are parallel.

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. Lines with slopes 3 and 3 are...

Choices: Parallel · Perpendicular · Neither · Vertical

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  1. Equal slopes are parallel.

Answer: Parallel

2. Lines with slopes 2 and -1/2 are...

Choices: Perpendicular · Parallel · Neither · Horizontal

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  1. They are opposite reciprocals.

Answer: Perpendicular

3. What slope is perpendicular to 4?

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  1. Use the opposite reciprocal.

Answer: -1/4

4. A quadrilateral has both pairs of opposite sides parallel. It is at least a...

Choices: Parallelogram · Triangle · Circle · Kite only

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  1. This is the definition of a parallelogram.

Answer: Parallelogram

5. A quadrilateral has four congruent sides. It is a...

Choices: Rhombus · Rectangle always · Circle · Triangle

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  1. A rhombus has four congruent sides.

Answer: Rhombus

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