This document defines properties of trapezoids and isosceles trapezoids. It provides examples of using these properties to solve problems. A trapezoid has one pair of parallel sides called bases and two nonparallel sides called legs. An isosceles trapezoid has congruent legs. The median of any trapezoid connects the midpoints of the legs and is parallel to the bases, with length equal to half the sum of the base lengths. Worked examples demonstrate using properties of trapezoids and isosceles trapezoids to find missing side lengths and angle measures.
2. A trapezoid is a quadrilateral (shape with four sides) with exactly one
pair of parallel sides. The parallel sides are also known as the bases of
the trapezoid. Whereas the nonparallel sides are called its legs. And the
angles formed by a base and the legs are called base angles.
AB & DC = base ∠A, ∠B, ∠C, & ∠D = base angles
AD & BC = legs
What is a trapezoid?
4. An isosceles trapezoid is a trapezoid whose legs (or
nonparallel sides) are congruent.
The properties of the isosceles trapezoid are as follows:
1. The base angles are congruent.
2. The diagonals are congruent.
What is an Isosceles Trapezoid?
5. Given: ABCD is an isosceles trapezoid
with bases AB and DC
Prove: TO ≅ TI
Proof:
Example #1:
6. Given: ZOID is an isosceles trapezoid
with bases OI and ZD.
Prove: TO ≅ TI
Proof:
Example #2:
Statements Reasons
1. Isosceles trapezoid ZOID Given
2. ZO ≅ DI Definition of isosceles trapezoid
3. ∠ZOI ≅ ∠DIO The upper base angles of the
isosceles trapezoid are
congruent.
4. OI ≅ IO Reflexive Property
5. ZOI ≅ DIO Side-Angle-Side or SAS theorem
6. ZOI ≅ DIO Corresponding parts of
congruent triangles are
congruent
7. TO ≅ TI Transitive property
7. Example #1: Use isosceles trapezoid DEGH to answer the following:
a. The legs in this trapezoid are line segment ED and line segment GH.
b. The bases in this trapezoid are line segment EG and line segment DH.
c. The base angles of trapezoid DEGH are ∠EDH and ∠GHD, ∠DEG and ∠HGE.
d. The diagonals in this trapezoid are line segment EH and line segment GD.
e. Line segment EH is the diagonals that are congruent to line segment GD
f. Line segment GH, which is the legs of a trapezoid.
g. ∠DEG is congruent to the other base angle, ∠HGE.
8. Example #2. In trapezoid PQRS, m∠R = 100 and m∠S = 5x + 35.
Find x and the measure of ∠S.
m∠R + m∠S = 180
100 + (5x + 35) = 180
5x + 135 = 180
5x = 180 - 135
5x/5 = 45/5
x = 9
m∠S = 5x + 35
m∠S = 5(9) + 35
= 45 + 35
= 80
Thus, m∠S = 80.
find x:
find the measure of ∠S:
9. Example #3: TUVS is an isosceles trapezoid.
If m∠U = 14x + 13 and m∠S = 13 + 8x, Find m∠V.
Solution: ∠T ≅ ∠U. ∠S and ∠T & ∠S and ∠U
are supplementary.
m∠S = 13 +8(7)
= 13 + 56
= 69
substitute x = 7 to find the measure of ∠S
Thus if ∠S ≅ ∠V, m∠V = 69.
V
10. The Median of a Trapezoid
The median (or the mid-segment) of a trapezoid is the line
segment joining or connecting the midpoints of the legs (or the
two non-parallel sides) of a trapezoid.
It is parallel to the bases of the trapezoid. The median of a
trapezoid is equal to one-half the sum of the lengths of the bases.
The Midline Theorem
The segment joining the midpoints of two non-parallel sides of a trapezoid
is parallel to the bases of the trapezoid and its length is one half the sum
of the measures of the bases.
11. Example #4: In figure 6.17, MN is the median of trapezoid STUR.
Solution:
a. If ST = 12 and RU = 22, then MN
a. If ST = 34 and MN = 40 then RU:
c. if RU = 34 and MN = 26 then ST:
d. If MN = 53, what is ST + RU?:
Thus, m∠R = 75 & m∠T = 108.
Remember: The length of the
median of a trapezoid is
equal to the average of the
lengths of the bases.
e. If m∠S = 105 and m∠U = 72,
find m∠R and m∠T:
∠S & ∠R = 180 (supplementary)
∠T & ∠U = 180 (supplementary)
12. Example #5: UT is the median ( or the mid-segment) of trapezoid JKLM.
a. If KL= 33, JM = x + 3, and UT = 4x - 10.
Find x, JM, and UT.
Solution:
● Find x: ● Substitute x = 8 to solve for
JM and UT.
Find JM: Find UT:
Thus, x = 8, segment JM = 11, and
segment UT = 22.
13. Example #5: UT is the median ( or the mid-segment) of trapezoid JKLM.
a. If KL = 33, JM = 3x - 6, and UT = x + 16.
Find KL, JM, and UT.
Solution:
● To find KL, JM, and UT, Find x first:
● Substitute x = 8 to solve for KL, JM, and UT.
14. • A trapezoid is a quadrilateral with only one pair of parallel sides.
• The parallel sides of a trapezoid are its bases and the nonparallel sides are its legs.
• The base angles of a trapezoid are the angles formed by a base and its legs.
• An isosceles trapezoid is a trapezoid whose legs are congruent.
• The diagonals of an isosceles trapezoid are congruent.
• The base angles of an isosceles trapezoid are congruent.
• The median of a trapezoid is the segment joining the midpoints of the legs.
• The length of the median of a trapezoid is equal to one-half the sum of the
lengths of the bases.
Some Key Points to Remember:
