1. Topic 6 Goals and Common Core Standards Ms. Helgeson
Identify, name, and describe polygons.
Use the sum of the measures of the interior angles of a quad.
Use some properties of parallelogram.
Prove that a quad is a parallelogram.
Use coordinate geometry with parallelograms

2. Use properties of sides and angles of rhombuses, rectangles, and squares
Use properties of diagonals of rhombuses, rectangles, and squares.
Use properties of trapezoids
Use properties of kites
Identify special quadrilaterals based on limited info
Prove that a quad is a special type of quad.

3. CC.9-12.G.CO.13
CC.9-12.G.CO.11
CC.9-12.G.SRT.5
CC.9-12.G.GPE.4
CC.9-12.G.CO.9

4. Quadrilaterals and Other Polygons
Topic 6
Quadrilaterals and Other Polygons

5. 6.1 Polygons
A polygon is a plane figure that meets the following conditions.
1. It is formed by three or more segments called sides, such that no two sides with a common endpoint are collinear.
2. Each side intersects exactly two other sides, one at each endpoint. Q R
vertex P
side S T
vertex

6. Identifying Polygons
POLYGON
NOT A POLYGON

7. POLYGON DEFINITION
A polygon is equilateral if all of its sides are congruent. A polygon is equiangular if all of its interior angles are congruent. A polygon is regular if it is equilateral and equiangular.

8. POLYGONS Number of Sides Type of Polygon 3 Triangle 4 Quadrilateral 5
Pentagon 6
Hexagon 7
Heptagon
Number of Sides
Type of Polygon 8
Octagon 9
Nonagon 10
Decagon 12
Dodecagon n
n-gon

9. interior
convex polygon
concave polygon or non-convex
A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that is not convex is called nonconvex or concave

10. A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Polygon PQRST has 2 diagonals from point Q, QT and QS.
diagonals

11. # of sides 3 4 5 6 n
diagonals 1 2
n - 3
triangles
n - 2
Sum of <‘s
180
360
540
720
180(n – 2)
There are n – 2 triangles in every n-sided polygon. Each triangle has an angle sum of 180˚.
Polygon Interior Angle-Sum Theorem:
Interior angle sum of an n-sided polygon = 180(n – 2)
Measure of an interior angle of a regular n-gon is
180(n – 2) n
The sum of the exterior angle measures of any convex polygon is 360˚.
Page 248 examples 4 and 5.

12. Theorem Interior Angles of a Quadrilateral
The sum of the measures of the interior angles of a quadrilateral is 360º.
m<1 + m<2 + m<3 + m<4 = 360º 4 1 3 2

13. Find m<U and m<V.
5xº
118º
(3x + 10)º
72º

14. Find m<D.
4xº
4xº
100º
2xº
2xº

16. Quadrilaterals
Isos. T.

17. 6.2 Trapezoids and Kites
What is the difference between a trapezoid and a kite?
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

18. Trapezoid (Not a Parallelogram)
4 sided figure.
Exactly one pair of opposite sides are parallel.
The median (midsegment) of a trapezoid is parallel to the bases, and its measure is ½ the sum of the measures of the bases.

19. Example: XY is the median of trap. MNOP
XY = 5k-7 MN = k+7 PO = 3k+3
K=___ MN=___ PO=___ XY=___
M N
X Y
P O

20. Example: Trap GHIJ with median KL.
GH= 3x-1 KL=10 JI= 7x+1
x=_____
G H
K L
J I

21. 3x + 2
2x + 4
2x + 1

22. Isosceles Trapezoid
4 sided figure, 1 pair of opposite sides are parallel. Other pair not parallel.
Legs are congruent.
Diagonals are congruent.
Both pairs of base angles are congruent.
base
base
angles
base
angles
leg
leg
base

23. Example: Isosceles Trap ABCD
AC = 7x BD = 5x+4 x=_____
D C
A B

24. Trapezoids B A E F G H C D
EF = ½ (AB – DC)
GE = FH

25. ABCD is a trapezoid with median MN.
If DC = 6 and AB = 16, find ME, FN, AND EF.
If DC = 3x, AB = 2x², and EF = 7, find the value of x. D C N M E F B A

26. Examples:
One angle of an isosceles trapezoid has measure 57. Find the measures of the other angles.
Two congruent angles of an isosceles trapezoid have measures 3x + 10 and 5x – 10. Find the value of x and and then give the measures of all angles of the trapezoid.

27. If AD = x and BE = x + 6, then x = ? And CF = ?.
TA = AB = BC AND TD = DE = EF
If AD = x and BE = x + 6, then x = ? And
CF = ?.
If AD = x + y, BE = 20, and CF = 4x – y, then CF = ? And y = ?.
If AD = x + 3, BE = x + y, and CF = 36, then x = ? And y = ? T A D E B F C

28. Kite
A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.
The diagonals are perpendicular.
Exactly one pair of opposite angles are congruent.

29. Kite C D B Seg. AC perp. to Seg. BD m<BAD = m<BCD; A
m<ABC ≠ m<ADC A

30. Example 1: GHJK is kite. Find HP.
√29 P 5 J G K

31. X + 30 X
125
Rstu IS A KITE. Find m< R, m< S, and m< T.

33. 6.3 Properties of Parallelograms
Both pairs of opposite sides are parallel. (Def)
Both pairs of opposite sides are congruent. (Thm)
Both pairs of opposite angles are congruent. (Thm)
Consecutive angles are supplementary. (Thm)
Diagonals bisect each other. (Thm)
One pair of sides are congruent and parallel (Thm)
Each diagonal divides quad. Into 2 congruent triangles.
4 sides figure - Quadrilateral

34. Example: Parallelogram TAXI
AX = 3y TI = 2y+10 TA = 3y+10
Find: AX=____ TI=____ TA=____
XI=____
m<R = x+20 m<T = 2x-30
Find: m<R=____ m<S=____
m<T=____ m<U=____
Example: Parallelogram RSTU

35. Example: Parallelogram RSTU
U T X
R S
RX = 3x-4 XT = 2x+4
Find: RX=___ XT=___ RT=___

37. 6.4 Proving Quad is a Parallelogram
A quadrilateral is a parallelogram if any one of the following is true.
Def: Both pairs of opposites are parallel.
Thm: Both pairs of opposites sides are congruent.
Thm: Both pairs of opposite angles are congruent.
Thm: Diagonals bisects each other.

38. 5. Thm: One pair of opposite sides are parallel and congruent
5. Thm: One pair of opposite sides are parallel and congruent. 6: Thm: Show that one angle is supplementary to both consecutive angles.

39. Slope Formula: m= y2-y1
x2-x1
Parallel lines: same slopes m1=m2
Perpendicular lines: neg. recip. m1m2=-1
Distance formula √(x2-x1)² + (y2-y1)²

41. 6.5 & 6.6 Rhombuses, Rectangles, and Squares

42. Rectangles (Parallelogram)
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
4 sided figure.
Contains 4 right angles.
Diagonals are congruent.

43. Example: Quadrilateral QUAD is a rectangle
Example: Quadrilateral QUAD is a rectangle. m<2 = 52, m<3 = 16x-12 m<2=____ m<3=____ m<1=____
Q U 2
m<2=_____________
m<3=_____________
m<1=_____________ 1 3
D A

44. Rhombus (Parallelogram)
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
4 sided figure.
All sides are congruent.
Diagonals bisect the opposite angles.
Diagonals are perpendicular.

45. Example: Rhombus BCDE m<1 = 2x+20 m<2 = 5x-4 x=____ m<1=____
Example: Use above figure.
m<3 = y²+26
y=____
x=____
m<1=____
m<2=____
m<3=____
B C 1 2 3 F
E D

46. Example: Rhombus WXYZ
m<XYZ=68 m<WXZ=____
Z Y
W X

47. Square (Parallelogram)
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
4 sided figure.
Contains 4 right angles.
All sides are congruent.
Diagonals bisect the angles.
Diagonals are congruent.
Diagonals are perpendicular.

48. 6.6 Special quadrilaterals (page 364)
Quadrilateral Kite Parallelogram Trapezoid Rhombus Rectangle Isos. Trap. Square