DWG NO. 06.4 — Lesson 4 of 5

Rectangles, Rhombi, and Squares

Unit 6: Quadrilaterals and Polygons · ~15–30 min

Objective

Identify rectangles, rhombi, and squares by their extra properties, and use diagonal relationships to solve for missing measures.

Special cases of the parallelogram

Rectangles, rhombi, and squares are all parallelograms — so every property from Lesson 6.2 still applies to each of them. What makes them special is an extra condition layered on top, which unlocks new facts about their diagonals.

Shape	Extra condition	Diagonal property
Rectangle	Four right angles	Diagonals are congruent
Rhombus	Four congruent sides	Diagonals are perpendicular and bisect the angles
Square	Four right angles and four congruent sides	Diagonals are congruent, perpendicular, and bisect the angles

A square is simply a rectangle and a rhombus at the same time, which is why it inherits every diagonal property from both.

Rectangle Diagonals Theorem: A parallelogram is a rectangle if and only if its diagonals are congruent.
Rhombus Diagonals Theorem: A parallelogram is a rhombus if and only if its diagonals are perpendicular.
Rhombus Angle Bisector Theorem: Each diagonal of a rhombus bisects the pair of opposite angles it connects.

In rhombus ABCD, diagonals AC and BD are perpendicular and each bisects a pair of opposite angles.

Worked Example 1 · Rectangle diagonal

ProblemRectangle RSTU has diagonal RT = 5x − 3 and diagonal SU = 2x + 12. Find x and the length of each diagonal.

1.In a rectangle, the diagonals are congruent: 5x − 3 = 2x + 12.

2.3x = 15, so x = 5. Then RT = 5(5) − 3 = 22, and SU = 22 as well.

x = 5, each diagonal = 22

Worked Example 2 · Rhombus angle from a diagonal

ProblemIn rhombus WXYZ, diagonal WY bisects ∠W, and m∠XWY = 34°. Find m∠W and m∠X.

1.Since the diagonal bisects ∠W into two equal halves, m∠W = 2(34°) = 68°.

2.∠W and ∠X are consecutive angles of a parallelogram, so they're supplementary: m∠X = 180° − 68° = 112°.

m∠W = 68°, m∠X = 112°

Guided practice

1.A rectangle has one diagonal of length 17. What is the length of the other diagonal?
Show answer
A rectangle's diagonals are congruent, so the other diagonal is also 17.
2.Rhombus DEFG has diagonals that intersect at H. What is m∠DHE?
Show answer
The diagonals of a rhombus are perpendicular, so m∠DHE = 90°.
3.A quadrilateral is both a rectangle and a rhombus. What shape must it be?
Show answer
A square — it has both four right angles and four congruent sides.
4.Rhombus PQRS has side length 6x − 5 and diagonal-created triangle side of 13. Given all four sides are congruent and one side equals 13, find x.
Show answer
All sides of a rhombus are congruent, so 6x − 5 = 13, giving 6x = 18, x = 3.
5.Explain why every square is a rhombus, but not every rhombus is a square.
Show answer
A square meets the rhombus condition (four congruent sides), so it automatically qualifies as a rhombus. But a rhombus only needs congruent sides — its angles don't have to be 90° — so a "squished" rhombus with slanted angles is still a rhombus without being a square.

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