Properties of Quadrilaterals

Parallelograms, rectangles, rhombi, squares, and trapezoids

Properties of Quadrilaterals

Parallelogram

A quadrilateral with both pairs of opposite sides parallel.

Properties:

  • Opposite sides are congruent
  • Opposite angles are congruent
  • Consecutive angles are supplementary
  • Diagonals bisect each other

Rectangle

A parallelogram with four right angles.

Additional properties:

  • All properties of parallelograms
  • Diagonals are congruent

Rhombus

A parallelogram with four congruent sides.

Additional properties:

  • All properties of parallelograms
  • Diagonals are perpendicular
  • Diagonals bisect the angles

Square

A parallelogram that is both a rectangle and a rhombus.

Properties:

  • Four congruent sides
  • Four right angles
  • Diagonals are congruent and perpendicular
  • Diagonals bisect the angles

Trapezoid

A quadrilateral with exactly one pair of parallel sides.

Parts:

  • Bases: the parallel sides
  • Legs: the non-parallel sides
  • Midsegment: connects midpoints of legs, length = b1+b22\frac{b_1 + b_2}{2}

Isosceles Trapezoid

A trapezoid with congruent legs.

Properties:

  • Base angles are congruent
  • Diagonals are congruent

📚 Practice Problems

1Problem 1easy

❓ Question:

In parallelogram ABCD, ∠A=65°\angle A = 65°. Find ∠C\angle C.

💡 Show Solution

In a parallelogram, opposite angles are congruent.

Since ∠A\angle A and ∠C\angle C are opposite: ∠C=∠A=65°\angle C = \angle A = 65°

Answer: ∠C=65°\angle C = 65°

2Problem 2medium

❓ Question:

A trapezoid has bases of length 8 and 14. Find the length of the midsegment.

💡 Show Solution

The midsegment of a trapezoid equals the average of the bases:

M=b1+b22M = \frac{b_1 + b_2}{2}

M=8+142=222=11M = \frac{8 + 14}{2} = \frac{22}{2} = 11

Answer: The midsegment is 11

3Problem 3hard

❓ Question:

In rhombus PQRS, diagonal PR = 16 and diagonal QS = 12. Find the length of one side of the rhombus.

💡 Show Solution

In a rhombus, diagonals are perpendicular and bisect each other.

The diagonals split the rhombus into 4 right triangles.

Half-diagonals:

  • Half of PR = 16/2=816/2 = 8
  • Half of QS = 12/2=612/2 = 6

Use Pythagorean Theorem: s2=82+62s^2 = 8^2 + 6^2 s2=64+36=100s^2 = 64 + 36 = 100 s=10s = 10

Answer: Each side is 10

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