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Parallelograms

Quadrilaterals are four-sided polygons

Parallelogram is a quadrilateral with both pairs

of opposite sides parallel.

Parallelograms (2)

A

B

D

C

Theorem 6.1 Opposite sides of a parallelograms

are congruent

AD ? BC and AB ? DC

Theorem 6.2 Opposite angles of a parallelogram

are congruent

ltA ? ltC and ltB ? ltD

mltAmltB 180 m ltBmltC 180 mltCmltD

180 mltDmltA 180

Theorem 6.3 Consecutive angles in a

parallelogram are supplementary.

Parallelograms (3)

Diagonals of a figure Segments that connect any

to vertices of a polygon

A

B

Theorem 6.4 The diagonals of a parallelogram

bisect each other.

D

C

Parallelograms (4)

- Draw a parallelogram ABCD on a piece

of construction paper. - Cut the parallelogram.
- Fold the paper and make a crease from A to C and

from B to D. - Fold the paper so A lies on C. What do you

observe? - Fold the paper so B lies on D. What do you

observe? - What theorem is confirmed by these Observations?

Tests for Parallelograms

Theorem 6.5 If both pairs of opposite sides of a

quadrilateral are congruent, then the

quadrilateral is a parallelogram.

A

B

If AD ? BC and AB ? DC, then ABCD is a

parallelogram

D

C

Theorem 6.6 If both pairs of opposite angles of

a quadrilateral are congruent, then the

quadrilateral is a parallelogram.

If ltA ? ltC and ltB ? ltD, then ABCD is a

parallelogram

Tests for Parallelograms 2

Theorem 6.7 If the diagonals of a quadrilateral

bisect each other, then the quadrilateral is a

parallelogram

A

B

D

C

Theorem 6.8 If one pair of opposite sides of a

quadrilateral is both parallel and congruent,

then the quadrilateral is a parallelogram.

A quadrilateral is a parallelogram if...

Both pairs of opposite sides are parallel.

(Definition)

Both pairs of opposite sides are congruent.

(Theorem 6.5)

Both pairs of opposite angles are congruent.

(Theorem 6.6)

Diagonals bisect each other. (Theorem 6.7)

A pair of opposite sides is both parallel and

congruent. (Theorem 6.8)

Area of a parallelogram

If a parallelogram has an area of A square units,

a base of b units and a height of h units, then A

bh. (Do example 1 p. 530)

h

b

The area of a region is the sum of the areas of

all its non-overlapping parts. (Do example 3 p.

531)

Rectangles

A rectangle is a quadrilateral with four right

angles.

Opp. angles in rectangles are congruent (they

are right angles) therefore rectangles are

parallelograms with all their properties.

Theorem 6-9 If a parallelogram is a rectangle,

then its diagonals are

congruent.

Theorem 6-10 If the diagonals of a

parallelogrma are congruent then the

parallelogram is a rectangle.

Rectangles (2)

- If a quadrilateral is a rectangle, then the

following properties hold true - Opp. Sides are congruent and parallel
- Opp. Angles are congruent
- Consecutive angles are supplementary
- Diagonals are congruent and bisect each other
- All four angles are right angles

Squares and Rhombi

A rhombus is a quadrilateral with four congruent

sides. Since opp. sides are ? , a rhombus is a

parallelogram with all its properties.

Special facts about rhombi Theorem 6.11 The

diagonals of a rhombus are

perpendicular. Theorem 6.12 If the diagonals of

a parallelogram are perpendicular, then

the parallelogram is a

rhombus. Theorem 6.13 Each diagonal of a rhombus

bisects a pair of opp.

angles

C

Squares and Rhombi(2)

If a rhombus has an area of A square units and

diagonals of d1 and d2 units, then A ½ d1d2.

If a quadrilateral is both, a rhombus and a

rectangle, is a square

Area of a triangle

h

b

If a triangle has an area of A square units a

base of b units and corresponding height of h

units, then A ½bh.

Congruent figures have equal areas.

Trapezoids

A trapezoid is a quadrilateral with exactly one

pair of parallel sides. The parallel sides are

called bases. The nonparallel sides are called

legs. At each side of a base there is a pair of

base angles.

C

Trapezoids (2)

A

B

AB base CD base AC leg BD leg

AB ?? CD AC BD are non parallel

C

D

ltA ltB pair of base angles ltC ltD pair of

base angles

C

Trapezoids (3)

Isosceles trapezoid A trapezoid with congruent

legs. Theorem 6-14 Both pairs of base angles of

an isosceles trapezoid are congruent. Theorem

6-15 The diagonals of an isosceles trapezoid are

congruent.

Trapezoids (4)

The median of a trapezoid is the segment that

joints the midpoints of the legs (PQ).

A

B

Q

P

C

D

Theorem 6-16 The median of a trapezoid is

parallel to the bases, and its measure is

one-half the sum of the measures of its bases.

C

Area of Trapezoids

B

A

h

C

D

Area of a trapezoid If a trapezoid has an area

of A square units, bases of b1 and b2 units and

height of h units, then A ½(b1 b2 )h.

C

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