MATH PROJECTAriella Lindenfeld & Nikki Colona

5.1 - PARALLELOGRAMS

A parallelogram is a quadrilateral with both pairs of opposite parallel sides.

Theorem 5-1: Opposite sides of the parallelogram are congruent.

Theorem 5-2: Opposite angles of a parallelogram are congruent.

Theorem 5-3: Diagonals of a parallelogram bisect each other.

G

F

H

E

EFGHEF = HG;FG=EH1

.1

2

3

4EFGH<3 = <4

3.

2. EFGH<E = <G

EXAMPLE:

X

120

y

7

38

X equals 120 degrees because of OAC which says that Opposite Angles are congruentY equals 22 because it supplementary to 120 So 60 minus 38 is 22

5.2 – PROVING QUADRILATERALS ARE PARALLELOGRAMS

Theorem 5-4: If both pairs of opposite sides of a quadrilateral are congruent , then the quadrilateral is a parallelogram.

Theorem 5-5: If one pair of opposite sides of a quadrilateral are both parallel, then the quadrilateral is a parallelogram.

Theorem 5-6: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 5-7: If the diagonals of a quadrilateral bisects each other, then the quadrilateral is a parallelogram.

T S

QR

1

23

4

<

<

<<

<<M

5.3 – THEOREMS INVOLVING PARALLEL LINES

Theorem 5-8: If two lines are parallel, then all points on the line are equidistant from the other line.

L

M

>

>

A B

C D

AB=CD

5.3 CONTINUED…

Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

AX II BY II CZ and AB = BCXY = YZ

A

B

C

X

Y

Z

1 2 3

4 5

SECTION 3 – THEOREMS INVOLVING PARALLEL LINES

Theorem 5-10

A line that contains the midpoint of one side of a triangle

and is parallel to another side passes through the

midpoint of the third side

Theorem 5-11

The segment that joins the midpoints of two sides of a

triangle

Is parallel to the third side

Is half as long as the third side

EXAMPLE:

5y+7

6y+3

4x+3

13x+1

2(4x+3) = 13x+18x+6=13x+15=5xX=1

5y+7 = 6y +34=y

X=1, Y=4

5-4 SPECIAL PARALLELOGRAMS

Rectangle is a quadrilateral with four right angles.

Therefore, every rectangle is a parallelogram.

Rhombus A quadrilateral with four congruent

sides.

Therefore, every rhombus is a parallelogram.

5-4 SPECIAL PARALLELOGRAMS

Square a quadrilateral with four right angles and four congruent sides. Therefore, every square is a rectangle, a rhombus, and

a parallelogram

*** since rectangles, rhombuses, and squares are parallelograms, they have all the properties of parallelograms.

5-4 SPECIAL PARALLELOGRAMS

Theorem 5-12

The diagonals of a rectangle are congruent

Theorem 5-13

The diagonals of a rhombus are perpendicular

Theorem 5-14

Each diagonal of a rhombus are perpendicular

5-4 SPECIAL PARALLELOGRAMS

Theorem 5-15

The midpoint of the hypotenuse of a right triangle is

equidistant from the three vertices

Theorem 5-16

If an angle of a parallelogram is a right angle, then the

parallelogram is a rectangle

Theorem 5-17

If two consecutive sides of a parallelogram are

congruent, then the parallelogram is a rhombus

5-5 TRAPEZOIDS

Trapezoid a quadrilateral with exactly one pair of

parallel sides

Bases the parallel sides

Legs the other sides

Isosceles Trapezoid a trapezoid with congruent

legs

Base angles are congruent

5-5 TRAPEZOIDS

Theorem 5-18

Base angles of an isosceles trapezoid are congruent

*** Median (of a trapezoid )the segment that joins the

midpoints of the legs

Theorem 5-19

The Median of a Trapezoid

Is parallel to the bases

Has a length equal to the average of the base lengths

PRACTICE QUESTIONS:

4x-y

95

7x-4

y1.

75

X z

y

2.

PRACTICE QUESTION

42

4x+y

26 3x-2y

3.

PROOF:

Statement Reason

1. Rectangle QRST RKST JQST

1.Given

KS =RTJT = QS

2. OSC

3. RT =QS 3. DB

4. JT = KS 4. substitution

T S

J Q R K