Mth 97

Winter

201

3

Sections 5.3 and 5.4

1

Section 5.3

–

Parallelograms and Rhombuses

Parallelograms

Theorem 5.14

–

A diagonal of a parallelogram forms two congruent triangles.

A

B

If

then

C

D

Corollary 5.15

–

In a parallelogram, the opposite sides are congruent and

the opposite angles are

congruent.

A

B

If

then

C

D

Corollary 5.16

–

Parallel lines are everywhere equidistant.

If

m

then

m

n

n

If

, then

Theorem 5.17

–

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is

a parallelogram.

Is the converse of 5.15?

A

B

If

then

Theorem 5.18

–

If both pairs of opposite angles of

a quadrilateral are congruent, then the quadrilateral

is a parallelogram.

A

B

If

then

A

B

C

D

D

C

D

C

Mth 97

Winter

201

3

Sections 5.3 and 5.4

2

Theorem 5.19

–

If a quadrilateral has two sides that are parallel and congruent, then it is a parallelogram.

A

B

If

then

C

D

Theorem 5.20

–

The diagonals of a parallelogram bisect each other.

A

B

A

B

If

M

then

C D

C

D

Theorem 5.21

–

If the diagonals of a quadrilateral bise

ct each other, then it is a parallelogram.

This theorem is the _______________________ of 5.20. Write the biconditional statement.

Rhombuses

E

Theorem 5.22

–

Every rhombus is a parallelogram.

Previous theorems ab

out parallelograms tell us…

H

M

F

1.

The diagonals of a rhombus bisect each other.

2.

The opposite angles of a rhombus are congruent. G

3.

The diagonals of a rhombus form two congruent triangles.

Theorem 5.23

–

The diagonal

s of a rhombus are perpendicular to each other.

A

If

D

B

then

C

Theorem 5.24

–

If the diagonals of a parallelogram are perpendicular to each other, then the

parallelogram is a rhombus

. Since this is the

converse

of theorem 5.23 write the biconditional statement.

Mth 97

Winter

201

3

Sections 5.3 and 5.4

3

Theorem 5.25

–

A parallelogram is a rhombus if and only if the diagonals bisect the opposite angles.

F

Write both statement

s

that this biconditional

statement says are true.

E

G

1)

H

2)

Do ICA 10

Section 5.4

–

Rectangles, Squares and Trapezoids

A

B

Rectangles

Theorem 5.26

–

Every rectangle is a parallelogram.

M

Previous theorems about parallelograms

tell us…

1.

The opposite sides are parallel.

D

C

2.

The opposite sides are congruent.

3.

The diagonals bisect each other.

Theorem 5.27

–

A parallelogram with one right angle is a rectangle.

E F

If EFGH is a pa

rallelogram and

ﮮ

F is a right angle,

then EFGH is a rectangle.

Given:

Prove: EFGH is a rectangle

H G

Subgoals: Prove

ﮮ

G is a right angle by

Prove

ﮮ

E is a right angle by

Prove

ﮮ

H is a right angle by

Write the inver

se of Theorem 5.27 and tell whether it is true or false.

Write the contrapositive of Theorem 5.27

and tell whether it is true or false.

Theorem 5.28

–

The diagonals of a rectangle are congruent.

S

T

If

then

V

U

Theorem 5.29

–

If a parallelogram has congruent diagonals, then it is a rectangle.

Is this the converse of Theorem 5.28?

Mth 97

Winter

201

3

Sections 5.3 and 5.4

4

Squares

Earlier we defined a square to be a quadrilateral with all sides congruent and four right angles.

Other possible definitions might be…

1.

A

square is a rhombus with one right angle.

A B

2.

A square is a rectangle with two adjacent sides congruent. D C

3.

A square is a rhombus with congruent diagonals.

4.

A square is a rectangle wit

h perpendicular diagonals.

Isosceles Trapezoids

E

F

An isosceles trapezoid has congruent legs.

H

G

Theorem 5.30

–

The base angles of an isosceles triangle are congruent.

If

EFGH is a trapezoid with

and _________________,

then

and _______________

See page 273 for proof.

Theorem 5.31 is the converse of Theorem 5.30. Write it below.

Write the inverse of Theorem 5.3

0 and tell whether it is true or false.

Write the contrapositive of Theorem 5.30

and tell whether it is true or false.

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