# 5.6 – Trapezoids

Key Terms

• Base Angles – The two angles formed by the base of a trapezoid and the two adjacent sides.
• Bases – The parallel sides of a trapezoid.
• Isosceles Trapezoid – A trapezoid with two congruent legs.
• The base angles of an isosceles trapezoid are also congruent.
• Legs – The nonparallel sides of a trapezoid.
• Median – The line segment that joins the midpoints of the legs of a trapezoid.
• The median is parallel to the bases, and its length equals the mean of their lengths.
• Trapezoid – A quadrilateral with exactly one pair of parallel sides.

Review

• Isosceles
• Two congruent sides
• Congruent base angles

• Consecutive Interior Angles
• Angles between two parallel lines, along the same side (transversal)
• Supplementary (adds up to 180°)

• Diagonals
• Segments that connect opposite vertices.

Notes

• Quadrilateral Family Tree and Diagrams

• Diagram of a Trapezoid

• Trapezoid Properties
• A trapezoid is a quadrilateral with exactly one pair of parallel sides.
• Trapezoids are NOT parallelograms
• In a parallelogram, both pairs of opposite sides are parallel.
• A trapezoid is not a parallelogram because only one pair of its sides is parallel.
• Trapezoids are NEVER rectangles, because they can only have ONE pair of parallel sides (not two).
• Trapezoids can have two right angles, but not four.

• Isosceles Trapezoids
• No right angles, not even one.

• Median
• … is a segment that connects the midpoints of the legs of a trapezoid.
• … is always parallel to both bases of a trapezoid.
• … Connects the midpoints of the legs.
• The length of the median is half the sum of the lengths of the bases.

• Median of a Trapezoid Formula
• $median=\frac{1}{2}\cdot (B_1+B_2)$

Examples

• Ex 1. If TRAP is an isosceles trapezoid, what is the value of x?
• Since $\overline{TR}$ and $\overline{PA}$ are parallel, $\angle T$ and $\angle P$ are consecutive interior angles (which are supplementary).
• Setup: 4x + 12 + 2x = 180
• Combine like terms: 6x + 12 = 180
• Subtract 12 on both sides: 6x = 168
• Divide by 6 on both sides: x = 28°

• Ex 2. If SLDG is an isosceles trapezoid, what is the value of x?
• Since the diagonals of an isosceles trapezoid are congruent, they are equal.
• Setup: 8x – 5 = 6x + 21
• Subtract 6x on both sides: 2x – 5 = 21
• Add 5 to both sides: 2x = 26
• Divide both sides by 2: x = 13°

•  Ex 3. Given the median $\overline{QR}$ and trapezoid MOPN, what is the value of x?
• The formula for finding the median is $median=\frac{1}{2}\cdot (B_1+B_2)$
• Setup: $\frac{1}{2}\cdot [(5x-7)+(6x+6)]=27$
• Combine like terms: $\frac{1}{2}\cdot (11x-1)=27$
• Multiply by 2 on both sides: 11x – 1 = 54
• Add 1 to both sides: 11x = 55
• Divide by 11 on both sides: x = 5

• Ex 4. DEFG is an isosceles trapezoid. Find the measure of $\angle G$.
• Isosceles trapezoids have congruent base angles. So, if $\angle D=121$ then $\angle G=121$
• Also, $\angle G$ and $\angle F$ are supplementary (add up to 180°); so, 59 + G = 180.  Solve for G.

• Ex 5. Jaime wants to tile his floor using tiles in the shape of a trapezoid.
• To make the pattern a little more interesting he has decided to cut the tiles in half along the median.
• The top base of each tile is 15 inches in length and the bottom base is 21 inches.
• How long of a cut will Jaime need to make so that he cuts the tiles along the median?
• Formula: $median=\frac{1}{2}\cdot (B_1+B_2)$
• Add the bases: $median=\frac{1}{2}\cdot (15+21)$
• What is one-half of 36?  Divide 36 by 2 to find out: $median=\frac{1}{2}\cdot (36)$
• The median = 18 inches