Rectangles, Rhombuses and Squares

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Rectangles, rhombuses and squares are special parallelograms. A rectangle has four right angles, and its diagonals are congruent. A rhombus has four congruent sides, and its diagonals are perpendicular and bisect its angles. A square is both a rectangle and a rhombus, so it has all of these properties.

Before this lesson: You should know Properties of Parallelograms, Proving a Quadrilateral Is a Parallelogram and the Distance Formula.

The Special Parallelograms

Every rectangle, rhombus and square is a parallelogram. So each one has all the parallelogram properties: opposite sides parallel and congruent, opposite angles congruent, and diagonals that bisect each other. Each special shape adds something more.

  • A rectangle is a parallelogram with four right angles.
  • A rhombus is a parallelogram with four congruent sides.
  • A square is a parallelogram with four right angles and four congruent sides.
A diagram of nested regions. The large outer region is labeled parallelograms. Inside it, two overlapping regions are labeled rectangles (four right angles) and rhombuses (four congruent sides). The overlap of rectangles and rhombuses is labeled squares.
Rectangles and rhombuses are both parallelograms. The shapes that are both are the squares.

Diagonal Properties

The biggest differences show up in the diagonals.

Property Parallelogram Rectangle Rhombus Square
Diagonals bisect each other Yes Yes Yes Yes
Diagonals are congruent No Yes No Yes
Diagonals are perpendicular No No Yes Yes
Diagonals bisect the angles No No Yes Yes
Four right angles No Yes No Yes
Four congruent sides No No Yes Yes
Three shapes side by side with their diagonals drawn. Rectangle: four right angles, and the two diagonals have matching marks on all four halves, showing they are congruent and bisect each other. Rhombus: four sides with one tick mark each, and the diagonals cross at a right angle and split each corner angle into two equal parts. Square: four equal sides, four right angles, and diagonals that are congruent and cross at a right angle.
A rectangle’s diagonals are congruent. A rhombus’s diagonals are perpendicular and split its angles in half. A square’s diagonals do both.

These also work as tests. A parallelogram with congruent diagonals is a rectangle. A parallelogram with perpendicular diagonals is a rhombus. A parallelogram with both is a square.

How to Solve Problems with Special Parallelograms

  1. Name the shape and list its properties from the table.
  2. Rectangle: set the two diagonals equal. Each half is half a diagonal.
  3. Rhombus: set sides equal, use 90° where the diagonals cross, and split each corner angle in half.
  4. Square: use all of the rectangle and rhombus facts. Each diagonal makes a 45° angle with the sides.
  5. Solve, then substitute back.

Worked Examples

Example 1: Diagonals of a rectangle

ABCD is a rectangle. Diagonal AC = 3x + 4 and diagonal BD = 5x - 10. Find x, AC and the length from A to the point E where the diagonals cross.

  1. The diagonals of a rectangle are congruent: 3x + 4 = 5x - 10
  2. Subtract 3x from both sides (see equations with variables on both sides): 4 = 2x - 10
  3. Add 10 to both sides: 14 = 2x
  4. Divide both sides by 2: 7 = x
  5. Find AC: 3(7) + 4 = 21 + 4 = 25
  6. Check with BD: 5(7) - 10 = 35 - 10 = 25
  7. The diagonals bisect each other, so AE is half of AC: 25 \div 2 = 12.5

Answer: x = 7, AC = 25 and AE = 12.5.

Example 2: Angles in a rhombus

ABCD is a rhombus, and its diagonals meet at E. m\angle ABD = 32^\circ. Find m\angle ABC, m\angle BAD, m\angle AEB and m\angle BAE.

  1. A rhombus’s diagonals bisect its angles, so BD splits angle B in half. Double it: 2 \times 32^\circ = 64^\circ. So m\angle ABC = 64^\circ.
  2. Consecutive angles add to 180°: 180^\circ - 64^\circ = 116^\circ. So m\angle BAD = 116^\circ.
  3. The diagonals are perpendicular: m\angle AEB = 90^\circ.
  4. In right triangle AEB, the acute angles add to 90° (see triangle angle sum theorem): 90^\circ - 32^\circ = 58^\circ.
  5. Check: AC bisects angle A, and 116^\circ \div 2 = 58^\circ. It matches.

Answer: m\angle ABC = 64^\circ, m\angle BAD = 116^\circ, m\angle AEB = 90^\circ and m\angle BAE = 58^\circ.

Example 3: Sides of a rhombus

ABCD is a rhombus with AB = 2y + 5 and BC = 4y - 3. Find y, the side length and the perimeter.

  1. All four sides of a rhombus are congruent: 2y + 5 = 4y - 3
  2. Subtract 2y from both sides: 5 = 2y - 3
  3. Add 3 to both sides: 8 = 2y
  4. Divide both sides by 2: 4 = y
  5. Find the side: 2(4) + 5 = 8 + 5 = 13
  6. Multiply by 4 for the perimeter: 4 \times 13 = 52

Answer: y = 4, each side is 13, and the perimeter is 52.

Example 4: Classify a shape on a grid

Quadrilateral ABCD has vertices A(-3, 0), B(0, 4), C(3, 0) and D(0, -4). Is it a rectangle, a rhombus, a square or none of these?

  1. Find AB (see distance formula): \sqrt{(0 + 3)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
  2. By the same steps, BC, CD and DA are also 5. All four sides are congruent, so ABCD is a rhombus.
  3. Diagonal AC runs from (−3, 0) to (3, 0), so AC = 6. Diagonal BD runs from (0, 4) to (0, −4), so BD = 8.
  4. The diagonals are not congruent (6 ≠ 8), so ABCD is not a rectangle.
  5. AC is horizontal and BD is vertical, so the diagonals are perpendicular, as a rhombus needs.

Answer: ABCD is a rhombus but not a rectangle, so it is not a square.

Example 5: Word problem with a gate frame

A carpenter builds a gate frame with top and bottom boards of 120 cm and side boards of 90 cm. Because opposite sides match, the frame is a parallelogram. She measures both diagonals and gets 150 cm each time. Is the frame a rectangle?

  1. The frame is a parallelogram, because both pairs of opposite sides are congruent.
  2. Both diagonals measure 150 cm, so the diagonals are congruent.
  3. A parallelogram with congruent diagonals is a rectangle.

Answer: Yes, the frame is a rectangle, so its corners are square. Builders use this diagonal check all the time.

Common Mistakes

Mistake Example of the error Fix
Giving every parallelogram congruent diagonals Setting the diagonals of a rhombus equal Only rectangles (and squares) have congruent diagonals.
Forgetting the diagonals of a rhombus are perpendicular Not using 90° where they cross In a rhombus, the diagonals always meet at a right angle.
Not halving the angle m\angle ABD = 32^\circ, so “m\angle ABC = 32^\circ” The diagonal splits the angle in half, so m\angle ABC = 64^\circ.
Thinking a square isn’t a rectangle “A square has equal sides, so it can’t be a rectangle” A square has four right angles, so it is a rectangle and a rhombus.
Using one diagonal fact to call a shape a square Congruent diagonals, so “square” Congruent diagonals give a rectangle. A square also needs perpendicular diagonals or equal sides.

Is Every Square a Rhombus?

  • Every square is a rectangle and a rhombus. Every rectangle and every rhombus is a parallelogram.
  • Not every rectangle is a square: a rectangle’s sides can have two different lengths.
  • Not every rhombus is a square: a rhombus can lean, with no right angles.
  • A kite also has perpendicular diagonals, but it is not a parallelogram. Kites come in the next lesson.

Practice Problems

  1. One diagonal of a rectangle is 18 cm. How long is the other diagonal?
  2. ABCD is a rectangle with diagonals AC = 4x - 2 and BD = 2x + 12. Find x and AC.
  3. A rhombus has a perimeter of 60 in. How long is each side?
  4. ABCD is a rhombus, and m\angle BAC = 35^\circ. Find m\angle BAD and m\angle ABC.
  5. The diagonals of a rhombus meet at E, and m\angle AEB = (5x + 15)^\circ. Find x.
  6. What angle does a diagonal of a square make with each side?
  7. A parallelogram has diagonals that are congruent and perpendicular. What is the most specific name for it?
  8. Show that A(1, 1), B(5, 1), C(5, 5) and D(1, 5) form a square.
  9. True or false: (a) Every square is a rhombus. (b) Every rhombus is a square. (c) Every rectangle is a parallelogram.
  10. A floor tile is a rhombus with one angle of 60°. Find its other three angles.
Show answers
  1. The diagonals of a rectangle are congruent
    The other diagonal is 18 cm.
  2. Set the diagonals equal: 4x - 2 = 2x + 12
    Subtract 2x: 2x - 2 = 12
    Add 2: 2x = 14
    Divide by 2: x = 7
    Find AC: 4(7) - 2 = 28 - 2 = 26
  3. All four sides are equal, so divide: 60 \div 4 = 15
    Each side is 15 in.
  4. AC bisects angle A: 2 \times 35 = 70, so m\angle BAD = 70^\circ
    Consecutive angles add to 180: 180 - 70 = 110, so m\angle ABC = 110^\circ
  5. The diagonals of a rhombus are perpendicular: 5x + 15 = 90
    Subtract 15: 5x = 75
    Divide by 5: x = 15
  6. The diagonal bisects a 90° corner: 90 \div 2 = 45
    It makes a 45° angle with each side.
  7. A square (congruent diagonals make a rectangle, perpendicular diagonals make a rhombus, and both make a square).
  8. Each side: AB is 5 - 1 = 4, BC is 5 - 1 = 4, CD is 5 - 1 = 4 and DA is 5 - 1 = 4
    AB is horizontal and BC is vertical, so angle B is a right angle (the same is true at each corner)
    Four congruent sides and four right angles make a square.
  9. (a) True (b) False (c) True
  10. Opposite angles are equal, so the angle across from 60° is 60°
    Consecutive angles add to 180: 180 - 60 = 120
    The angles are 60°, 120°, 60° and 120°.

FAQ

What is the difference between a rhombus and a square?

Both have four congruent sides. A square also has four right angles, while a rhombus can have angles that aren’t 90°.

Are the diagonals of a rectangle perpendicular?

Only if the rectangle is a square. In other rectangles, the diagonals are congruent but do not meet at right angles.

Is a square a rectangle?

Yes. A square has four right angles, which is all a rectangle needs. A square is also a rhombus.

How do you prove a parallelogram is a rectangle?

Show that it has one right angle, or that its diagonals are congruent. Either fact makes it a rectangle.

Summary

  • Rectangles, rhombuses and squares are special parallelograms.
  • A rectangle has four right angles and congruent diagonals.
  • A rhombus has four congruent sides, and its diagonals are perpendicular and bisect its angles.
  • A square has all of these properties.
  • A parallelogram with congruent diagonals is a rectangle; with perpendicular diagonals, it is a rhombus.

Previous lesson: Proving a Quadrilateral Is a Parallelogram

Next lesson: Trapezoids and Kites

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