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Proving a quadrilateral is a specific type

20 practice questions 2 video lessons Theory + worked examples

Classifying Quadrilaterals by Proof

Texas Geometry (TEKS) • Standard G.6(E) • Quadrilateral Theorems

Classifying Quadrilaterals by Proof is a topic in Quadrilateral Theorems in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.6(E), which requires students to prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using its properties.

Proving a quadrilateral's type uses its side, angle, and diagonal properties, often with coordinates, to show it is a parallelogram, rectangle, rhombus, or square.

Texas Geometry (TEKS) › Quadrilateral Theorems › Classifying Quadrilaterals by Proof  —  Standard G.6(E)

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Practice questions

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Watch 2 video(s)
  • How to Prove That a Quadrilateral Is a Parallelogram With Diagonals : Parallelograms & Math Watch
  • Coordinate Proofs (Geometry) Proving a Quadrilateral is a Parallelogram (4 Ways) Watch
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Theory

To prove a quadrilateral's type, use coordinate geometry to test its sides and diagonals:

  • Parallel sides — equal slopes.
  • Perpendicular (right angle) — slopes multiply to \(-1\).
  • Congruent sides/diagonals — equal distances (distance formula).

Build up the classification: parallelogram \(\to\) add a right angle for a rectangle, or equal adjacent sides for a rhombus, or both for a square.

Prove the general type first, then the special feature. A rectangle is a parallelogram plus a right angle.
Proving a quadrilateral type on the plane Coordinates let you prove a quadrilateral's type by testing side slopes and lengths. A B C D
Plot the vertices, then test sides with slope and distance.
Coordinate tests Coordinate tests Coordinate tests parallel: equal slopes perpendicular: slopes multiply to −1 congruent: equal distances
The three coordinate tests.

The coordinate tools:

\[\text{slope}=\dfrac{y_2-y_1}{x_2-x_1},\qquad d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]
slope is rise over run; distance is the square root of the sum of squared differences
Parallel \(\Rightarrow\) equal slopes; perpendicular \(\Rightarrow\) slopes with product \(-1\).

How to run a coordinate proof

  1. Compute the slopes of the four sides to check for parallel and perpendicular pairs.
  2. Compute distances to check for congruent sides or diagonals.
  3. Conclude the most specific type the evidence supports.
Example 1 — Prove a parallelogram
How do you prove a quadrilateral is a parallelogram using slopes?
Solution

Show both pairs of opposite sides are parallel by checking that each pair has equal slope.

\(\text{slope } AB\)\(=\)\(\text{slope } DC\)
\(\text{slope } BC\)\(=\)\(\text{slope } AD\)
show both pairs of opposite sides have equal slopes
Example 2 — Prove a rectangle
After proving a parallelogram, how do you show it is a rectangle?
Solution

Show one angle is a right angle: two adjacent sides are perpendicular, so their slopes multiply to \(-1\).

\((\text{slope } AB)(\text{slope } BC)\)\(=\)\(-1\)
show adjacent sides are perpendicular, slopes multiply to negative one
Example 3 — Prove a rhombus
How do you show a parallelogram is a rhombus with coordinates?
Solution

Show two adjacent sides are congruent using the distance formula — then all four sides are equal.

\(AB\)\(=\)\(BC\ \text{(distance formula)}\)
show two adjacent sides are equal by the distance formula
Example 4 — Classify
A quadrilateral has both pairs of opposite sides parallel and equal, perpendicular adjacent sides, but unequal adjacent sides. What is it?
Solution

Parallel opposite sides make it a parallelogram; a right angle makes it a rectangle; unequal adjacent sides mean it is not a square. So it is a rectangle.

it is a rectangle

Common pitfalls

Equal slopes prove parallel; the distance formula proves congruent. Use the right tool for each claim.
A right angle needs slopes with product \(-1\), not just different slopes.
Prove enough for the specific type. A parallelogram alone isn't a rectangle without a right angle.

Frequently asked questions

How do you prove a quadrilateral is a parallelogram with coordinates?

Show both pairs of opposite sides are parallel (equal slopes), or both pairs congruent (equal distances).

How do you prove a rectangle using coordinates?

Prove it is a parallelogram, then show one angle is right: two adjacent sides have slopes whose product is \(-1\).

How do you prove a rhombus using coordinates?

Prove it is a parallelogram, then show two adjacent sides are equal using the distance formula.

What tells you two lines are perpendicular from their slopes?

Their slopes multiply to \(-1\) (they are negative reciprocals).

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