Proving Properties of Special Quadrilaterals - Classroom...

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Proving Properties of Special QuadrilateralsAdapted from Walch Education

Rectangle

• A rectangle has four sides and four right angles.

• A rectangle is a parallelogram, so opposite sides are parallel, opposite angles are congruent, and consecutive angles are supplementary.

• The diagonals of a rectangle bisect each other and are also congruent.

Rectangle

Theorem

If a parallelogram is a rectangle, then the diagonals are congruent.

AC @ DB

Rhombus

• A rhombus is a special parallelogram with all four sides congruent.

• Since a rhombus is a parallelogram, opposite sides are parallel, opposite angles are congruent, and consecutive angles are supplementary.

• The diagonals bisect each other; additionally, they also bisect the opposite pairs of angles within the rhombus.

• The diagonals of a rhombus also form four right angles where they intersect.

Rhombus

Theorem

If a parallelogram is a rhombus, the diagonals of the rhombus

bisect the opposite pairs of angles.

ÐBAC @ ÐCAD @ ÐBCA @ ÐDCA

ÐCBD @ ÐABD @ ÐADB @ ÐCDB

Rhombus, continued

Theorem

If a parallelogram is a rhombus, the diagonals are

perpendicular.

The converse is also true. If the diagonals of a parallelogram

intersect at a right angle, then the parallelogram is a rhombus.

BD ^ AC

Square

• A square has all the properties of a rectangle and a rhombus.

• Squares have four congruent sides and four right angles.

• The diagonals of a square bisect each other, are congruent,

and bisect opposite pairs of angles.

• The diagonals are also perpendicular.

Square

Properties of Squares

AB @ BC @ CD @ DA

BD @ AC

BD ^ AC

mÐA = mÐB = mÐC =

mÐD = 90

Trapezoid

• Trapezoids are quadrilaterals with exactly one pair of opposite

parallel lines.

• Trapezoids are not parallelograms because they do not have

two pairs of opposite lines that are parallel.

• The lines in a trapezoid that are parallel are called the bases,

and the lines that are not parallel are called the legs.

Trapezoid

Properties of Trapezoids

and are the legs.

and are the bases.

Isosceles Trapezoid

• Isosceles trapezoids have one pair of opposite parallel lines.

The legs are congruent.

• Since the legs are congruent, both pairs of base angles are

also congruent, similar to the legs and base angles in an

isosceles triangle.

• The diagonals of an isosceles trapezoid are congruent.

Isosceles Trapezoid

Properties of Isosceles Trapezoids

and are the legs.

and are the bases.

BA @ CD and AC @ BD

ÐBAD @ ÐADC and

ÐABC @ ÐBCD

• A kite is a quadrilateral with two distinct pairs of congruent

sides that are adjacent.

• Kites are not parallelograms because opposite sides are not

parallel.

• The diagonals of a kite are perpendicular.

Properties of Kites

CD @ CB and AB @ AD

CA ^ BD

Hierarchy of Quadrilaterals

Practice

• Quadrilateral ABCD has vertices A (–6, 8), B (2, 2), C (–1, –2), and D (–9, 4). Using slope, distance, and/or midpoints, classify as a rectangle, rhombus, square, trapezoid, isosceles trapezoid, or kite.

Graph the Quadrilateral

Calculate the slopes of the sides

If opposite sides are parallel, the quadrilateral is a parallelogram.

The first pair of opposite sides is parallel:

(2 - 8)

[2 - (-6)]=

(-2 - 4)

[-1- (-9)]=

Calculate the slopes of the sides

The second pair of opposite sides is parallel:

Therefore, the quadrilateral is a parallelogram.

(4 - 8)

[-9 - (-6)]=

(-2 - 2)

(-1- 2)=

Examine the slopes of the consecutive sides

If the slopes are opposite reciprocals, the lines are perpendicular and therefore form right angles. If there are four right angles, the quadrilateral is a rectangle or a square.

Examine the slopes of the consecutive sides

is the opposite reciprocal of .

The slopes of the consecutive sides are perpendicular: and .

• There are four right angles at the vertices.

• The parallelogram is a rectangle or a square.

AB ^ AD DC ^ BC

Determining whether the diagonals are congruent

• If the diagonals are congruent, then the parallelogram is a

rectangle or square.

• determine if the diagonals are congruent by calculating the

length of each diagonal using the distance formula,

d = (x

1)2 + (y

The parallelogram is a rectangle.

• The diagonals are congruent: .

AC = [-1- (-6)]2 + (-2 - 8)2

AC = (5)2 + (-10)2

AC = 25 +100

AC = 125

AC = 5 5

AC @ DB

DB = [2 - (-9)]2 + (2 - 4)2

DB = (11)2 + (-2)2

DB = 121+ 4

DB = 125

DB = 5 5

Calculate the length of the sides

• If all sides are congruent, the parallelogram is a rhombus or a square.

• Since we established that the angles are right angles, the rectangle can be more precisely classified as a square if the sides are congruent.

• If the sides are not congruent, the parallelogram is a rectangle.

• Use the distance formula to calculate the lengths of the sides. 1

AB = [2 - (-6)]2 + (2 - 8)2

AB = (8)2 + (-6)2

AB = 64 + 36

AB = 100

AB = 10

AD = [-9 - (-6)]2 + (4 - 8)2

AD = (–3)2 + (-4)2

AD = 9 +16

AD = 25

AD = 5

d = (x

1)2 + (y

• Opposite sides are congruent, which is consistent with a parallelogram, but all sides are not congruent.

DC = [-1- (-9)]2 + (-2 - 4)2

DC = (8)2 + (-6)2

DC = 64 + 36

DC = 100

DC = 10

BC = (-1- 2)2 + (-2 - 2)2

BC = (–3)2 + (-4)2

BC = 9 +16

BC = 25

BC = 5

Summarizing the findings

The quadrilateral has opposite sides that are parallel and four right angles, but not four congruent sides. This makes the quadrilateral a parallelogram and a rectangle.

Try this one…

• Quadrilateral ABCD has vertices A (0, 8), B (11, 1), C (0, –6), and D (–11, 1). Using slope, distance, and/or midpoints, classify as a rectangle, rhombus, square, trapezoid, isosceles trapezoid, or kite.

THANKS FOR WATCHING!

Dr. Dambreville

Proving Properties of Special Quadrilaterals - Classroom...

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