Quadrilaterals are four-sided polygons. There are all kinds of quadrilaterals. Squares and rectangles are common quadrilaterals. Trapezoids, parallelograms, and rhombuses are also quadrilaterals. Quadrilaterals are everywhere. A playing card, a window frame, and index cards are all quadrilaterals.
A trapezoid is a quadrilateral with exactly one pair of parallel sides. Both of these figures are trapezoids.
In both figures, 
is parallel to 
. Notice that 
is not parallel to BD.
The parallel sides of the trapezoid are called the bases of the trapezoid. and are the bases of these trapezoids.
The nonparallel sides of the trapezoid are called the legs of the trapezoid. and 
are the legs of these trapezoids.
The median of a trapezoid is a line segment that is parallel to both bases of the trapezoid and connects both legs of the trapezoid at their midpoints. In trapezoid ABCD, 
is the median of the trapezoid. The length of the median of a trapezoid is
EXAMPLE:
If one base of a trapezoid is 6, and the other base is 12 inches, what is the length of the median?
Here is a picture of a trapezoid. is the median of the trapezoid.
What do you know about this trapezoid?
and are the bases of the trapezoid.
and 
are the legs of the trapezoid.
, , and are all parallel to each other.
≅ 
.
≅ 
.
Do this experiment to find some common properties of trapezoids.
Experiment
Explore the properties of trapezoids.
Materials
Paper
Pencil
Scissors
Tape
Procedure
1.On a piece of paper, draw three large triangles. Draw one obtuse triangle, one right triangle, and one isosceles triangle.
2.Cut out all three of these triangles.
3.Draw a line parallel to the base of each of the triangles.
4.Cut off the top of the triangles on the line. The result is three trapezoids.
5.Rip off the angles on the same side of the trapezoid. Tape both of these angles together. Together they should equal 180 degrees.
6.Rip off all four angles from one of the trapezoids. Tape all four of the angles together. What is the sum of all four angles?
Something to think about . . .
If you draw a line on a trapezoid parallel to the base of the trapezoid, two new shapes result. What are they?
If you draw a line down the center of a trapezoid perpendicular to the base of the trapezoid, two new shapes result. What are they?
Angle A and angle C are supplementary.
m∠A + m∠C = 180
Angle B and angle D are supplementary.
m∠B + m∠D = 180
Theorem: The sum of the angles of a trapezoid is 360 degrees.
m∠A + m∠B + m∠C + m∠D = 360
MINI-PROOF
Prove that the sum of the angles of a trapezoid is 360 degrees.
First list what you know.
1.ABDC is a trapezoid.
2.Angles A and C are supplementary since both angles on the same side of a trapezoid are supplementary.
3.Angles B and D are supplementary since both angles on the same side of a trapezoid are supplementary.
4.m∠A + m∠C = 180 since supplementary angles total 180 degrees.
5.m∠B + m∠D = 180 since supplementary angles total 180 degrees.
Conclusion
m∠A + m∠B + m∠C + m∠D = 360 degrees
An isosceles trapezoid is a special type of trapezoid, where both legs are congruent. To draw an isosceles trapezoid, start by drawing an isosceles triangle. Draw a line parallel to the base of the triangle. Cut off the top of the triangle along the line you just drew. The result is an isosceles trapezoid.
An isosceles trapezoid has two pairs of base angles.
Angles A and B are one pair of base angles of the trapezoid.
Angles C and D are the other pair of base angles of the trapezoid.
The base angles of an isosceles trapezoid are congruent.
∠A ≅ ∠B.
∠C ≅ ∠D.
The legs of an isosceles triangle are congruent.
≅ .
The diagonals of an isosceles trapezoid are congruent.
≅ .
Experiment
Discover the sum of the angles of an isosceles trapezoid.
Materials
Paper
Pencil
Procedure
1.Draw an isosceles trapezoid.
2.Draw a single diagonal in the trapezoid. This diagonal divides the trapezoid into two triangles.
3.Label the triangles triangle 1 and triangle 2.
4.What is the measure of the sum of the angles of triangle 1?
5.What is the measure of the sum of the angles of triangle 2?
6.Add the sum of the angles of these two triangles together.
180 + 180 = 360
The total is 360 degrees.
Something to think about . . .
Is there another way to divide an isosceles trapezoid into two equal shapes?
The sum of the angles of an isosceles trapezoid is 360 degrees.
m∠A + m∠B + m∠C + m∠D = 360
Theorem: The opposite angles of an isosceles trapezoid are supplementary.
EXAMPLE:
Angles 1 and 4 are supplementary.
m∠1 + m∠4 = 180
Angles 2 and 3 are supplementary.
m∠2 + m∠3 = 180
Compute the values of the angles of this isosceles trapezoid.
1.If m∠2 = 30, what is the m∠5?
2.If m∠1 = 30, what is the m∠4?
Compute the measures of the angles for the isosceles trapezoid if m∠3 is 120 degrees.
3.What is the m∠1?
4.What is the m∠2?
5.What is the m∠4?
6.What is the m∠5?
7.What is the m∠6?
(Answers)
A parallelogram is a quadrilateral with two pairs of parallel sides. All these figures are parallelograms.
In all these figures, side AB is parallel to side CD, and side AC is parallel to side BD.
Theorem: Opposite angles of a parallelogram are congruent.
∠A ≅ ∠D
∠C ≅ ∠B
Theorem: The opposite sides of a parallelogram are congruent.
EXAMPLE:
AB ≅ CD
AC ≅ BD
EXAMPLE:
m∠1 + m∠2 = 180
m∠2 + m∠3 = 180
m∠3 + m∠4 = 180
m∠4 + m∠1 = 180
Experiment
Explore the relationship between the angles of a parallelogram.
Materials
Protractor
Pencil
Paper
Procedure
1.Draw three different parallelograms.
2.Label the angles of each parallelogram angles 1, 2, 3, and 4.
3.Measure each of the angles of the parallelogram. Enter the measurements in the chart.
4.Add the measurements of the pairs of angles indicated in the chart. Enter the results in the chart.
5.Add the measurements of all four angles together. Enter the results in the chart.
Something to think about . . .
What did you notice about the patterns formed?
Can you draw a parallelogram where the sum of the angles is not 360 degrees?
Theorem: The sum of the angles of a parallelogram is 360 degrees.
EXAMPLE:
m∠1 + m∠2 + m∠3 + m∠4 = 360
Theorem: Each diagonal of a parallelogram separates the parallelogram into two congruent triangles.
EXAMPLE:
Triangle ABC ≅ Triangle BCD
Triangle ABD ≅ Triangle ACD
MINI-PROOF
How can you prove triangle ABC is congruent to triangle CAD?
First list what you know.
1.ABCD is a parallelogram.
2. is a diagonal of parallelogram ABCD.
What does this tell you?
3. ≅ since every segment is congruent to itself.
4. ≅ since opposite sides of a parallelogram are congruent.
5. ≅ since opposite sides of a parallelogram are congruent.
Conclusion
Triangle ABC is congruent to triangle CAD since three sides of one triangle are congruent to three sides of another triangle.
Theorem: The diagonals of a parallelogram bisect each other.
Diagonals AD and BC intersect at point E.
=
= 
Experiment
Compare the diagonals of a parallelogram.
Materials
Pencil
Paper
Scissors
Procedure
1.Draw a parallelogram. Connect points A and D. Call this parallelogram 1.
2.Copy the parallelogram. Connect points B and C. Call this parallelogram 2.
3.Cut parallelogram 1 on diagonal AD.
4.Cut parallelogram 2 on diagonal BC.
5.Compare the length of diagonal AD to diagonal BC. Are they both the same length?
Something to think about . . .
Can you draw a parallelogram where the diagonals are equal? What shape is it?
This figure is a parallelogram. The m∠5 is 110.
1.What is m∠1?
2.What is m∠2?
3.What is m∠3?
4.What is m∠4?
5.What is m∠6?
Find the measure of the indicated angles in this parallelogram when m∠5 = 30, m∠6 = 40, and m∠7 = 50.
6.What is m∠1?
7.What is m∠2?
8.What is m∠8?
9.What is m∠11?
10.What is m∠12?
(Answers)
A rhombus is a parallelogram with four equal sides. Since it is a parallelogram, opposite sides of a rhombus are parallel. All these shapes are rhombuses.
Theorem: The diagonals of a rhombus bisect the angles of the rhombus.
EXAMPLE:
Diagonal bisects ∠CAB and ∠BDC.
The m∠1 = m∠2.
The m∠7 = m∠8.
Diagonal bisects ∠ABD and ∠DCA.
The m∠3 = m∠4.
The m∠5 = m∠6.
Theorem: The diagonals of a rhombus are perpendicular.
These diagonals intersect at point E.
is perpendicular to .
∠AEB is a right angle.
∠BEC is a right angle.
∠CED is a right angle.
∠DEA is a right angle.
Since a rhombus is a special parallelogram with four equal sides, everything that applies to parallelograms also applies to rhombuses.
•The opposite angles of a rhombus are congruent.
•The consecutive pairs of angles of a rhombus are supplementary.
•The diagonals of a rhombus separate the rhombus into two congruent triangles.
•The diagonals of a rhombus bisect each other.
This figure is a rhombus. The measure of ∠5 is 40. Determine the measures of the indicated angles.
1.∠ACD
2.∠4
3.∠1
4.∠9
5.∠11
(Answers)
A rectangle is a parallelogram with four right angles. Both figures are rectangles.
m∠2 = 90
m∠3 = 90
m∠4 = 90
Theorem: All four angles of a rectangle are congruent.
EXAMPLE:
Angle A = Angle B = Angle C = Angle D
Theorem: Any two angles of a rectangle are supplementary.
m∠1 + m∠2 = 180 degrees; ∠1 and ∠2 are supplementary.
m∠1 + m∠3 = 180 degrees; ∠1 and ∠3 are supplementary.
m∠1 + m∠4 = 180 degrees; ∠1 and ∠4 are supplementary.
m∠2 + m∠3 = 180 degrees; ∠2 and ∠3 are supplementary.
m∠2 + m∠4 = 180 degrees; ∠2 and ∠4 are supplementary.
m∠3 + m∠4 = 180 degrees; ∠3 and ∠4 are supplementary.
Since a rectangle is a parallelogram, the theorems that apply to parallelograms also apply to rectangles.
•The diagonals of a rectangle separate the rectangle into two congruent triangles.
•The diagonals of a rectangle bisect each other.
This figure is a rectangle, and m∠1 = 30.
1.What is the measure of angle 2?
2.What is the measure of angle 3?
3.What is the measure of angle 4?
4.What is the measure of angle 5?
5.What is the measure of angle 6?
(Answers)
A square is a rectangle with four equal sides.
Since a square is a type of rectangle, the measure of all the angles of a square is 90 degrees.
m∠A = 90
m∠B = 90
m∠C = 90
m∠D = 90
Theorem: All four sides of a square are congruent.
EXAMPLE:
= = = 
Look at the relationship between the diagonals of a square.
Theorem: The diagonals of a square are perpendicular to each other.
Diagonal is perpendicular to diagonal .
Angle AEC is a right angle.
Angle AEB is a right angle.
Angle BED is a right angle.
Angle CED is a right angle.
Since a square is a rectangle and a rectangle is a type of parallelogram, all the theorems that apply to parallelograms and to rectangles also apply to squares.
•The diagonals of squares are equal.
•The measures of all the angles of a square are equal.
•The sum of the angles of a square is 360 degrees.
•The diagonals of a square bisect each other.
•The triangles formed by the diagonals of a square are congruent.
Discover the relationship between the four small triangles formed by the diagonals of a square.
Materials
Paper
Pencils
Scissors
Procedure
1.Draw a square.
2.Draw the diagonals of the square.
3.Cut the square along the diagonals and form four small triangles.
4.Place these four triangles on top of each other. Are they congruent?
5.What other shapes can you construct from these four small triangles?
Something to think about . . .
Do the diagonals of a rectangle form four identical triangles?
Determine the relationship between the following pairs of angles contained in the square.
C = Complementary angles
S = Supplementary angles
E = Equal angles
V = Vertical angles
A = Adjacent angles
1.∠CAB and ∠BDC
2.∠1 and ∠2
3.∠1 and ∠4
4.∠3 and ∠6
5.∠1 and ∠5
(Answers)
Decide what type of quadrilateral could be represented by the following statements. Circle all that apply.
T represents Trapezoid
P represents Parallelogram
RH represents Rhombus
R represents Rectangle
S represents Square
T P RH R S
T P RH R S
T P RH R S
T P RH R S
T P RH R S
T P RH R S
T P RH R S
1.Two pairs of parallel sides.
2.All four angles are right angles.
3.Diagonals are perpendicular.
4.Only two sides are parallel.
5.All the angles can be different.
6.Diagonals are equal.
7.Opposite angles are congruent.
8.Opposite angles are supplementary.
(Answers)
