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Geometry

Quadrilateral Properties

Lesson

Special quadrilaterals inherit properties as you specialize: parallelogram → rectangle / rhombus → square.

Parallelogram

  • Opposite sides are parallel and equal.
  • Opposite angles equal.
  • Consecutive angles sum to 180°.
  • Diagonals bisect each other.

Rectangle

A parallelogram with four right angles. Diagonals are equal.

Rhombus

A parallelogram with four equal sides. Diagonals are perpendicular and bisect the angles.

Square

A rectangle AND a rhombus. Everything!

Trapezoid

Exactly one pair of parallel sides. Area: 12(b1+b2)h\frac{1}{2}(b_1 + b_2) h.

Kite

Two pairs of adjacent equal sides. Area: 12d1d2\frac{1}{2} d_1 d_2 (same formula as rhombus).

Worked example

Trapezoid bases 5 and 9, height 4:

A=12(5+9)(4)=28A = \tfrac{1}{2}(5 + 9)(4) = 28

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Parallelogram angle 70∘. Opposite angle?\text{Parallelogram angle } 70^\circ. \text{ Opposite angle?}

Problem 2

Parallelogram angle 70∘. Adjacent angle?\text{Parallelogram angle } 70^\circ. \text{ Adjacent angle?}

Problem 3

Rectangle diagonal length 12. Other diagonal?\text{Rectangle diagonal length 12. Other diagonal?}

Problem 4

Square side 6. Perimeter?\text{Square side 6. Perimeter?}

Practice

Standard problems matching the lesson.

Problem 5

Parallelogram angles 50∘ and x. Find adjacent x.\text{Parallelogram angles } 50^\circ \text{ and } x. \text{ Find adjacent } x.

Problem 6

Parallelogram sides 5 and 8. Perimeter?\text{Parallelogram sides 5 and 8. Perimeter?}

Problem 7

Rectangle 4 by 9. Perimeter?\text{Rectangle 4 by 9. Perimeter?}

Problem 8

Rectangle 4 by 9. Area?\text{Rectangle 4 by 9. Area?}

Problem 9

Rhombus side 7. Perimeter?\text{Rhombus side 7. Perimeter?}

Problem 10

Square diagonal 82. Side?\text{Square diagonal } 8\sqrt{2}. \text{ Side?}

Problem 11

Rhombus diagonals 6 and 8. Area?\text{Rhombus diagonals 6 and 8. Area?}

Problem 12

Trapezoid bases 5, 9, height 4. Area?\text{Trapezoid bases 5, 9, height 4. Area?}

Problem 13

Sum of interior angles in any quadrilateral\text{Sum of interior angles in any quadrilateral}

Problem 14

Consecutive angles in a parallelogram sum to?\text{Consecutive angles in a parallelogram sum to?}

Problem 15

Kite diagonals 10 and 6. Area?\text{Kite diagonals 10 and 6. Area?}

Problem 16

Rectangle diagonal 13, sides 5 and 12. Area?\text{Rectangle diagonal 13, sides 5 and 12. Area?}

Problem 17

Square perimeter 40. Side?\text{Square perimeter 40. Side?}

Problem 18

Three angles of quad: 90∘,80∘,70∘. Fourth?\text{Three angles of quad: } 90^\circ, 80^\circ, 70^\circ. \text{ Fourth?}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Isosceles trapezoid base angle 65∘. Other base angle?\text{Isosceles trapezoid base angle } 65^\circ. \text{ Other base angle?}

Problem 20

Rhombus side 13, one diagonal 24. Other diagonal?\text{Rhombus side 13, one diagonal 24. Other diagonal?}

Problem 21

Parallelogram perim 32, one side 6. Other?\text{Parallelogram perim 32, one side 6. Other?}

Problem 22

Square area 64. Diagonal coefficient (⋅2)\text{Square area 64. Diagonal coefficient }(\cdot \sqrt{2})

Problem 23

Trapezoid parallel sides 7, 13, area 60. Height?\text{Trapezoid parallel sides 7, 13, area 60. Height?}

Problem 24

Rectangle perim 28, width 5. Length?\text{Rectangle perim 28, width 5. Length?}

Problem 25

Rhombus side 5, one diagonal 6. Area?\text{Rhombus side 5, one diagonal 6. Area?}

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