What is a Parallelogram?
A parallelogram is a quadrilateral with both pairs of opposite sides that are parallel.
The Properties of a Parallelogram
- The opposite sides are of equal length in a parallelogram.
- The opposite angles are of equal measure.
- The consecutive angles of a parallelogram are supplementary.
- The diagonals bisect each other in a parallelogram.
- The diagonal of a parallelogram divides it into two triangles that are congruent to each other.
What is a Trapezoid?
A trapezoid is a quadrilateral with exactly one pair of opposite parallel sides.
The Properties of a Trapezoid
- The pairs of parallel sides are called bases.
- The non-parallel sides are called legs.
- The two consecutive angles whose common side is the base are called base angles.
- The angles formed between the parallel sides are supplementary.
- If the legs of a trapezoid are congruent, it is called an isosceles trapezoid.
What is a Kite?
A kite is a quadrilateral whose two pairs of adjacent sides are congruent but whose opposite sides are not congruent.
The Properties of a Kite
- The diagonals are perpendicular to each other in the kite.
- The angles where the unequal sides meet are equal in measure.
- The longer diagonal bisects the other diagonal.
Solved Examples
Example 1: A playground is in the shape of a parallelogram. One of its side 20 yards, as shown in the figure below. The side opposite to it is x + 5 yards. Find the value of x.
Solution:
Let the playground which is a parallelogram be represented by ABCD,
AB = CD The opposite sides of a parallelogram are equal
x + 5 = 20 Substitute
x – 5 = 20 – 5 Subtract 5 from each side
x = 15
So, the value of x is 15 yards.
Example 2: Find the measure of \(\angle\)BCD if the quadrilateral ABCD is an isosceles trapezoid.
Solution:
ABCD is an isosceles trapezoid.
\(\angle\)ADC = \(\angle\)BCD The base angles of an isosceles trapezoid are equal
70° = \(\angle\)BCD Substitute
So, the measure of BCD is 70°.
Example 3: Find the value of y if the given quadrilateral is a kite.
Solution:
ABCD is a kite
AO = CO The longer diagonal bisects the other diagonal
y – 3 = 7 Substitute
y – 3 + 3 = 7 + 3 Add 3 to each side
y = 10
So, the value of y is 10.
Example 4: Find the measure of \(\angle\)ABC if the given quadrilateral is a parallelogram.
Solution:
ABCD is a parallelogram, use the parallelogram opposite angles theorem.
\(\angle\)ABC = \(\angle\)ADC The opposite angles of a parallelogram are equal
\(\angle\)ABC = 75° Substitute
So, the measure of \(\angle\)ABC is 75°.
A parallelogram has two pairs of parallel sides. In a trapezoid, however, only one pair of the opposite sides is parallel.
All the properties of a trapezoid are present in a parallelogram, so all parallelograms are trapezoids.
A rectangle and a square are two quadrilaterals with each angle measuring 90°.
A trapezoid whose legs are of equal measure is an isosceles trapezoid.
There are three types of parallelograms:
- Rhombus
- Rectangle
- Square