How to Understand the Key Properties of Trapezoids

Step-by-step Guide: Properties of Trapezoids

Defining a Trapezoid:
A trapezoid (or trapezium in some regions) is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, while the non-parallel sides are the legs.

Base Angles:

• Isosceles Trapezoid: A trapezoid in which the non-parallel sides (legs) are congruent. In this type of trapezoid, base angles are congruent.
• Non-Isosceles Trapezoid: Base angles are not necessarily congruent.

Midsegment (Median) of a Trapezoid:
The segment that joins the midpoints of the legs of a trapezoid is called the midsegment. This segment is parallel to the bases and its length is the average of the lengths of the bases.

Parallel Sides and Angles:
In any trapezoid, the interior angles on the same side of the legs sum to \(180^\circ\).

Height and Area:
The height (or altitude) of a trapezoid is the perpendicular distance between its parallel sides (bases).
The area, \(A\), of a trapezoid is given by:
\( A = \frac{1}{2} (b_1 + b_2) \times h\)
where \(b_1\) and \(b_2\) are the lengths of the bases and \(h\) is the height.

Examples

Example 1: Midsegment of a Trapezoid
In trapezoid \( ABCD \) with \( AD \) parallel to \( BC \), if \( AD \) measures \( 6 \) units and \( BC \) measures \( 10 \) units, determine the length of the midsegment.

Solution:
To start, remember the definition of the midsegment in a trapezoid: It connects the midpoints of the two non-parallel sides (legs) and its length is the average of the two bases.
Using the given measurements, we can apply the formula:
\( \text{Midsegment} = \frac{\text{Base 1} + \text{Base 2}}{2} = \frac{AD + BC}{2} = \frac{6 + 10 }{2} = \frac{16 }{2} = 8 \)
The midsegment of trapezoid \(ABCD\) is \(8\) units in length.

Example 2: Area of a Trapezoid
Determine the area of trapezoid \( PQRS \), where \( PQ \) is parallel to \( SR \), \( PQ \) measures \( 5 \) units, \( SR \) measures \( 9 \) units, and the height of the trapezoid is \( 4 \) units.

Solution:
To determine the area of a trapezoid, we utilize the formula:
\( A = \frac{1}{2} (b_1 + b_2) \times h \)
Where \( A \) is the area, \( b_1 \) and \( b_2 \) are the lengths of the bases, and \( h \) is the height.
Plugging in our given values:
\( A = \frac{1}{2} (PQ + SR) \times \text{height} = \frac{1}{2} (5 + 9) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 \)
The area of trapezoid PQRS is \( 28 \) square units.

Example 3: Isosceles Trapezoid Angles
In an isosceles trapezoid, if one of the base angles measures \(60^\circ\), what is the measure of the angle adjacent to it?

Solution:
An isosceles trapezoid has congruent legs, and its base angles (angles adjacent to the same base) are congruent.
If one base angle is given as \(60^\circ\), then the adjacent base angle, which is congruent to it, will also be \(60^\circ\).
In an isosceles trapezoid, if one base angle measures \(60^\circ\), the adjacent base angle also measures \(60^\circ\).

Practice Questions:

1. If one pair of opposite sides in a quadrilateral are parallel and the other pair is non-parallel, what type of quadrilateral is it?
2. In a trapezoid, if the non-parallel sides are of equal length, what special name is given to this trapezoid?
3. True or False: The diagonals of an isosceles trapezoid are always equal in length.
4. In a trapezoid \(ABCD\) where \(AB\) is parallel to \(CD\), if angle \(A\) measures \(75^\circ\), what would be the measure of angle \(C\) given that the angles on the same side of the non-parallel line are supplementary?

Answer:

1. Trapezoid.
2. Isosceles trapezoid.
3. True.
4. \(105^\circ\) (since \(180^\circ – 75^\circ = 105^\circ\)).