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All courses Math II · G-CO.11 35 of 73
Prove parallelogram theorems and their converses, including diagonal and rectangle results.

Use the opposite-sides theorem in a parallelogram to find a missing side measure or solve for a variable

Problem
In parallelogram ABCD, AB=12. Find the missing side measure.
Unworked quadrilateral configuration showing only the supplied evidence. Open full size
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Hint

Use the parallelogram property that opposite sides have equal length.

In parallelogram \(ABCD\), side \(AB\) is opposite side \(CD\), so \(AB=CD\).

Solution walkthrough

01

Verify the opposite-side pairing

\[\begin{aligned} \text{boundary}~\text{order}~A-B-C-D \\ AB~\text{is}~\text{opposite}~CD \end{aligned}\]

Sides AB and CD share no endpoint and lie across the parallelogram from one another, as the inspected labels and matching marks show.

02

Apply the parallelogram side theorem

\[AB=CD\]

Opposite sides of a parallelogram are congruent.

03

Substitute the known length

\[AB=12~->~CD=12\]

The given length transfers directly to its opposite side.

04

Check the named pair

\[AB=CD=12\]

The equality uses the correct one-mark opposite pair, not the adjacent double-mark pair AD and BC.

+

Another way

  1. Use a diagonal to form congruent triangles and recover the opposite-side equality.

!

Common mistake

Do not use an adjacent side. The congruent partner of AB is the side directly opposite it, CD.

Completed quadrilateral diagram with decisive theorem evidence and conclusion.
CD = 12