1. Quadrilaterals and Polygons
Polygon: A plane figure that is formed by three or more segments (no two of which are collinear), and each segment (side) intersects at exactly two other sides – one at each endpoint (Vertex).
Which of the following diagrams are polygons?

2. # of Sides Type of Polygon 3 4 5 6 7 # of Sides Type of Polygon 8 9 10
Polygons are Named & Classified by the Number of Sides They Have
# of Sides
Type of Polygon 3 4 5 6 7
# of Sides
Type of Polygon 8 9 10 12 #
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodagon
N-gon
What type of polygons are the following?

3. Convex and Concave Polygons
Convex – A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon.
Concave – A polygon that is not convex
Interior
Interior
Equilateral, Equiangular, and Regular

4. Diagonals and Interior Angles of a Quadrilateral
Diagonal – a segment that connects to non-consecutive vertices.
Theorem 6.1 – Interior Angles of a Quadrilateral Theorem
The sum of the measures of the interior angles of a quadrilateral is 360O
m<1 + m<2 + m<3 + m<4 = 360o
80o o
xo xo

5. Properties of Parallelograms
Theorem 6.2
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
PQ = RS and SP = QR
Theorem 6.3
If a quadrilateral is a parallelogram, then it opposite angles are congruent.
<P = < R and < Q = < S
Theorem 6.4
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
m<P + m<Q = 180o, m<Q + m<R = 180o
m<R + m<S = 180o, m<S + m<P = 180o
Theorem 6.5
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
QM = SM and PM = RM _ ~
Q R
P S

6. Using the Properties of Parallelograms
FGHJ is a parallelogram.
Find the length of:
a. JH
b. JK
F G K
J H
PQRS is a parallelogram.
Find the angle measures:
a. m<R
b. m<Q
Q R
70o
P S
Find the value of x
P Q
3xo o
S R

7. Proofs Involving Parallelograms
A E B 2 1
D C 3
G F
Given: ABCD and AEFG are
parallelograms
Prove: <1 = < 3
Statements Reasons ~
ABCD & AEFG are Parallelograms Given
<1 = < Opposite Angles are congruent (6.3)
<2 = < Opposite Angles are Congruent (6.3)
<1 = < Transitive Property of Congruence ~
Plan: Show that both angles are congruent to <2

8. Given: ABCD is a parallelogram Prove: AB = CD, AD = CB
Proving Theorem 6.2
A B
D C
Given: ABCD is a parallelogram
Prove: AB = CD, AD = CB
Statements Reasons ~ _
ABCD is a parallelogram 1. Given
Draw Diagonal BD 2. Through any two points there
exists exactly one line
3. AB || CD, and AD || CB 3. Def. of a parallelogram __
4. <ABD = < CDB Alternate Interior Angles Theorem
5. <ADB = < CBD Alternate Interior Angles Theorem
6. DB = DB Reflexive Property of Congruence
7. /\ ADB = /\ CBD ASA Congruence Postulate
8. AB = CD, AD = CB 8. CPCTC __ ~
Plan: Insert a diagonal which will allow us to divide the parallelogram into two triangles

9. Proving Quadrilaterals are Parallelograms
Theorem 6.6
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Q R
P S
Q R
P S
Theorem 6.7
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Q R
P S
(180-x)o xo xo
Theorem 6.8
If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram
Q R
P S
Theorem 6.9
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

10. Concept Summary – Proving Quadrilaterals are Parallelograms
Show that both pairs of opposite sides are
Show that both pairs of opposite angles are
Show that one angle is supplementary to
Show that the diagonals
Show that one pair of opposite sides are both
congruent
parallel
BOTH consecutive interior <‘s
bisect each other
congruent and ||

11. Proving Quadrilaterals are Parallelograms – Coordinate Geometry
B(1,3)
D (7,1)
A(2, -1)
How can we prove that the Quad is a parallelogram?
1. Slope - Opposite Sides ||
2. Length (Distance Formula) – Opposite sides same length
3. Combination – Show One pair of opposite sides both || and congruent

12. Rhombuses, Rectangles, and Squares
Square – a parallelogram with four congruent sides and four right angles
Rhombus – a parallelogram with four congruent sides
Rectangle – a parallelogram with four right angles
Parallelograms
Rhombuses
Rectangles
Squares

13. Using Properties of Special Triangles
If ABCD is a rectangle, what else do you know about ABCD?
A B
C D
Corollaries about Special Quadrilaterals
Rhombus Corollary – A quad is a rhombus if and only if it has four congruent sides
Rectangle Corollary – A quad is a rectangle if and only if it has four right angles
Square Corollary – A quad is a square if and only if it is a rhombus and a rectangle
How can we use these special properties and corollaries of a Rhombus?
P Q
S R
2y + 3
5y - 6

14. Theorem 6.11: Theorem 6.12: Theorem 6.13:
Using Diagonals of Special Parallelograms
Theorem 6.11:
A parallelogram is a rhombus if and only if its diagonals are perpendicular
Theorem 6.12:
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
Theorem 6.13:
A parallelogram is a rectangle if and only if its diagonals are congruent

15. Decide if the statement is sometimes, always, or never true.
A rhombus is equilateral.
2. The diagonals of a rectangle are _|_.
3. The opposite angles of a rhombus are supplementary.
4. A square is a rectangle.
5. The diagonals of a rectangle bisect each other.
6. The consecutive angles of a square are supplementary.
Always
Sometimes
Quadrilateral ABCD is Rhombus.
7. If m <BAE = 32o, find m<ECD.
8. If m<EDC = 43o, find m<CBA.
9. If m<EAB = 57o, find m<ADC.
10. If m<BEC = (3x -15)o, solve for x.
11. If m<ADE = ((5x – 8)o and m<CBE = (3x +24)o, solve for x
12. If m<BAD = (4x + 14)o and m<ABC = (2x + 10)o, solve for x.
A B E
D C
32o
86o
66o
35o 16 26

16. Coordinate Proofs Using the Properties of Rhombuses, Rectangles and Squares
Using the distance formula and slope, how can we determine the specific shape of a parallelogram?
Rhombus –
Rectangle –
Square -
Based on the following Coordinate values, determine if each parallelogram
is a rhombus, a rectangle, or square.
P (-2, 3) P(-4, 0)
Q(-2, -4) Q(3, 7)
R(2, -4) R(6, 4)
S(2, 3) S(-1, -3)

17. Given: HIJK is a parallelogram /\ HOI = /\ JOI
Prove: HIJK is a Rhombus
Statements Reasons
H I O
K J ~

18. Given: RECT is a Rectangle Prove: /\ ART = /\ ACE Statements Reasons ~
T C

19. Given: PQRT is a Rhombus Prove: PR bisects <TPQ and < QRT,
and QT bisects <PTR and <PQP
Statements Reasons
P Q
T R
Plan: First prove that Triangle PRQ is congruent to Triangle PRT; and Triangle TPQ is congruent to Triangle TRQ

20. Trapezoids and Kites
A B
D C
>
A Trapezoid is a Quad with exactly one pair of parallel sides. The parallel sides are the BASES. A Trapezoid has exactly two pairs of BASE ANGLES
In trapezoid ABCD, Which 2 sides are the bases? The legs? Name the pairs of base angles.
A B
D C
>
If the legs of the trapezoid are congruent, then the trapezoid is an Isosceles Trapezoid.

21. Theorems of Trapezoids
If a trapezoid is isosceles, then each pair of base angles is congruent.
<A = <B = <C = <D ~
A B
D C
>
Theorem 6.15
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
ABCD is an isosceles trapezoid
A B
D C
> ~
Theorem 6.16
A trapezoid is isosceles if and only if its diagonals are congruent.
ABCD is isosceles if and only if AC = BD _
A B
D C
>

22. Kites and Theorems about Kites
A kite is a quadrilateral that has two pairs of consecutive congruent sides,
But opposite sides are NOT congruent.
Theorem 6.18
If a Quad is a Kite, then its diagonals are perpendicular.
Theorem 6.19
If a Quad is a kite then exactly one pair of opposite angles are congruent

23. Using the Properties of a KiteX 12 U
W Y Z
Find the length of WX, XY, YZ, and WZ. J
H 132o o K G
Find the angle measures of <HJK and < HGK

24. Summarizing the Properties of Quadrilaterals
______________ _________________ ________________
____________ _____________ ____________ ______________
Kites Parallelograms Trapezoids
Rhombus Squares Rectangles Isosceles Trap.

25. Properties of Quadrilaterals
X X X X X
X X X
X X
X X X X
X X X X
X X
X X X X

26. Area of a Square Postulate
Using Area Formulas
Area of a Square Postulate
The area of a square is the square of the length of its side.
Area Congruence Postulate
If two polygons are congruent then they have the same area.
Area Addition Postulate
The area of a region is the sum of the area of its non-overlapping sides.
Area of a Rectangle
The area of a rectangle is the product of its base and height.
A = bh
Area of a Parallelogram
The area of a parallelogram is the product of a base, and it’s corresponding height
Area of a Triangle
The area of a triangle is one half the product of a base and its corresponding height
A = ½bh h b

28. Name That Proof

29. Prove: MRON is a parallelogram Statements Reasons ~ __
Given: /\ RQP = /\ ONP
R is the midpoint of MQ
Prove: MRON is a parallelogram
Statements Reasons ~ __ Q
R P O
M N
1. /\ RQP = /\ ONP Given
R is the midpoint of MQ
2. MR = RQ Definition of a midpoint
3. RQ = NO CPCTC
4. MR = NO Transitive Property of Congruency
5. <QRP = < NOP CPCTC
6. MQ || NO Alternate Interior <‘s Converse
7. MRON is a parallelogram Theorem 6.10 ~ __

30. Given: UWXZ is a parallelogram, <1 = <8
Prove: UVXY is a parallelogram
Statements Reasons
U V W
Z Y X 1 8 ~
1. UWXZ is a parallelogram Given
2. UW || ZX Definition of a parallelogram
3. UV || YX Segments of Congruent Segments
4. <Z = <W Opposite <‘s of a parallelogram are =
5. <1 = < Given
6. <5 = < Third Angles Theorem
6. <4 = < Alternate Interior Angles Theorem
6. <5 = < Transitive Property of Congruence
7. UY || VX Corresponding Angles Converse
8. UVXY is a parallelogram Definition of a Parallelogram ~ __

31. Given: GIJL is a parallelogram Prove: HIKL is a parallelogram
Statements Reasons
L K J M
G H I
GIJL is a parallelogram Given
GI || LJ Definition of a parallelogram
<GIL = <JLI Alternate Interior Angles Theorem
GJ Bisects LI Diagonals of a parallelogram bisect
MI = ML Definition of a Segment Bisector
<HMI = <KML Vertical Angles Theorem
/\ HMI = /\ KML ASA Congruence Postulate
MH = MK CPCTC
HK and IL Bisect Each other Definition of a Segment Bisector
HIKL is a parallelogram Theorem 6.9 __ ~