1. Geometry 1 Unit 6
Quadrilaterals

2. Geometry 1 Unit 6
6.1 Polygons

3. Polygons Polygon Sides Vertex A closed figure in a plane
Formed by connecting line segments endpoint to endpoint
Each segment intersects exactly 2 others
Classified by the number of sides they have
Named by listing vertices in consecutive order
Sides
Line segments in a polygon
Vertex
Each endpoint in a polygon

4. Polygons
polygons not polygons

5. Polygons
Pentagon ABCDE or pentagon CDEAB A D C B E

6. Polygons Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
11 Undecagon
12 Dodecagon
n n-gon (a 19 sided polygon is a 19-gon)

7. Polygons
Diagonals
Line segments that connect non-consecutive vertices.

8. Polygons Convex polygons
Polygons with no diagonals on the outside of the polygon

9. Polygons Concave polygons
A polygon is concave if at least one diagonal is outside the polygon
These are also called nonconvex.

10. Polygons
Example 1
Identify the polygon and state whether it is convex or concave.

11. Polygons Equilateral Polygon Equiangular Polygon Regular Polygon
all sides the same length
Equiangular Polygon
all angles equal measure
Regular Polygon
equilateral and equiangular

12. Polygons
Equilateral Equiangular Regular

13. Polygons
Example 2
Decide whether the polygon is regular.

14. Polygons Interior Angles of a Quadrilateral Theorem
The sum of the measures of the interior angles of a quadrilateral is 360°.
m1 + m2 + m3 + m4 = 360° 1 4 3 2

15. Polygons
Example 3
Find mF, mG, and mH. x
55° E H F G

16. Polygons Example 4 Use the information in the diagram to solve for x
100°
2x + 30
3x – 5
120°

17. 6.2 Properties of Parallelograms
Geometry 1 Unit 6
6.2 Properties of Parallelograms

18. Properties of Parallelograms
Quadrilateral with two pairs of parallel sides.

19. Properties of Parallelograms
Opposite Sides of a Parallelogram Theorem
If a quadrilateral is a parallelogram, then its opposite sides are congruent. P Q S R
PQ  RS and SP  QR

20. Properties of Parallelograms
Opposite Angles in a Parallelogram Theorem
If a quadrilateral is a parallelogram, then its opposite angles are congruent. P Q S R
P  R and Q  S

21. Properties of Parallelograms
Consecutive Angles in a Parallelogram Theorem
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary R
Add to equal 180° Q
mP + mQ = 180°
mQ + mR = 180°
mR + mS = 180°
mS + mP = 180° S P

22. Properties of Parallelograms
Diagonals in a Parallelogram Theorem
If a quadrilateral is a parallelogram, then its diagonals bisect each other. P Q S R M
QM  SM and PM  RM

23. Properties of Parallelograms
Example 1
GHJK is a parallelogram. Find each unknown length JH LH G K H J L 6 8  
JH = 8
LH = 6

24. Properties of Parallelograms
Example 2
In ABCD, mC = 105°. Find the measure of each angle.
mA
mD

25. Properties of Parallelograms
Example 3
WXYZ is a parallelogram. Find the value of x. Z Y 
3x + 18°  
4x – 9°  W X

26. Properties of Parallelograms
Statement
Reasons
ABCD is a parallelogram
AD || BC
2  1
AB || CD
Alternate interior angles theorem
2  4
Example 4
Given:
ABCD is a parallelogram.
Prove:
2  4 A D B C 1 4 2 3

27. Properties of Parallelograms
Statement
Reason
ACDF is a parallelogram.
ABDE is a parallelogram.
Opposite sides of a parallelogram are congruent
AC = DF
AB = DE
AC = AB + BC
DF = DE + EF
AC = DE + DF
AB + BC = AB + EF
BC = EF
Def of Congruent
∆BCD  ∆EFA
Example 5
Given:
ACDF is a parallelogram.
ABDE is a parallelogram.
Prove:
∆BCD  ∆EFA A B C F D E

28. Properties of Parallelograms
Example 6
A four-sided concrete slab has consecutive angle measures of 85°, 94°, 85°, and 96°. Is the slab a parallelogram? Explain.

29. 6.3 Proving Quadrilaterals are Parallelograms
Geometry 1 Unit 6
6.3 Proving Quadrilaterals are Parallelograms

30. Proving Quadrilaterals are Parallelograms
Investigating Properties of Parallelograms
Cut 4 straws to form two congruent pairs.
Partly unbend two paperclips, link their smaller ends, and insert the larger ends into two cut straws. Join the rest of the straws to form a quadrilateral with opposite sides congruent.
Change the angles of your quadrilateral. Is your quadrilateral a parallelogram?

31. Proving Quadrilaterals are Parallelograms
Converse of the Opposite Sides of a Parallelogram Theorem
If a opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. D A C B
ABCD is a parallelogram

32. Proving Quadrilaterals are Parallelograms
Converse of the Opposite Angles in a Parallelogram Theorem
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. D A C B
ABCD is a parallelogram.

33. Proving Quadrilaterals are Parallelograms
Converse of the Consecutive Angles in a Parallelogram Theorem
If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. D A C B
(180 – x)° x°
ABCD is a parallelogram

34. Proving Quadrilaterals are Parallelograms
Converse of the Diagonals in a Parallelogram Theorem
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. D A C B M
ABCD is a parallelogram

35. Proving Quadrilaterals are Parallelograms
Example 1
Given:
∆PQT  ∆RST
Prove:
PQRS is a parallelogram.
Statements
Reasons
∆PQT  ∆RST
CPCTC
PT = RT
ST = QT
Def. of bisect
PQRS is a parallelogram P Q R S T

36. Proving Quadrilaterals are Parallelograms
Example 2
A gate is braced as shown. How do you know that opposite sides of the gate are congruent?

37. Proving Quadrilaterals are Parallelograms
Congruent and Parallel Sides Theorem
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. A B D C 
ABCD is a parallelogram.

38. Proving Quadrilaterals are Parallelograms
To determine if a quadrilateral is a parallelogram, you need to know one of the following:
Opposite sides are parallel
Opposite sides are congruent
Opposite angles are congruent
An angle is supplementary with both of its consecutive angles
Diagonals bisect each other
One pair of sides is both parallel and congruent

39. Proving Quadrilaterals are Parallelograms
Example 3
Show that A(-1,2), B(3,2), C(1,-2), and D(-3,-2) are the vertices of a parallelogram.

40. 6.4 Rhombuses, Rectangles, and Squares
Geometry 1 Unit 6
6.4 Rhombuses, Rectangles, and Squares

41. Rhombuses, Rectangles, and Squares
Parallelogram with four congruent angles
Rhombus
Parallelogram with four congruent sides
Square
Parallelogram with four congruent angles and four congruent sides

44. Rhombuses, Rectangles, and Squares
Example 1
Decide if each statement is always, sometimes or never true.
A rhombus is a rectangle
A parallelogram is a rectangle
A rectangle is a square
A square is a rhombus

45. Rhombuses, Rectangles, and Squares
Example 2
Given FROG is a rectangle, what else do you know about FROG? F R G O

46. Rhombuses, Rectangles, and Squares
Example 3
EFGH is a rectangle. K is the midpoint of FH. EG = 8z – 16,
What is the measure of segment EK?
What is the measure of segment GK?

47. Rhombuses, Rectangles, and Squares
Rhombus Corollary
A quadrilateral is a rhombus if and only if it has four congruent sides.

48. Rhombuses, Rectangles, and Squares
Rectangle Corollary
A quadrilateral is a rectangle if and only if it has four right angles.

49. Rhombuses, Rectangles, and Squares
Square Corollary
A quadrilateral is a square if and only if it is a rhombus and a rectangle.

50. Rhombuses, Rectangles, and Squares
Perpendicular Diagonals of a Rhombus Theorem
A parallelogram is a rhombus if and only if its diagonals are perpendicular.   B A C D
ABCD is a rhombus if and only if
AC BD.

51. Rhombuses, Rectangles, and Squares
Diagonals Bisecting Opposite Angles Theorem.
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
ABCD us a rhombus if and only if
AC bisects DAB and BCD
and
BD bisects ADC and CBA A B D C

52. Rhombuses, Rectangles, and Squares
Diagonals in a Rectangle Theorem
A parallelogram is a rectangle if and only if its diagonals are congruent. A B D C
ABCD is a rectangle if and only if
AC  BD.

53. Rhombuses, Rectangles, and Squares
Example 4
You cut out a parallelogram shaped quilt piece and measure the diagonals to be congruent. What is the shape?
An angle formed by the diagonals of the quilt piece measures 90°. Is the shape a square?
Rectangle
Yes

54. Geometry 1 Unit 6
6.5 Trapezoids and Kites

55. Trapezoids and Kites Trapezoid Bases Pairs of Base Angles Legs
A quadrilateral with exactly one pair of parallel sides.
Bases
The parallel sides of a trapezoid.
Pairs of Base Angles
Angles in a trapezoid that share a base.
Legs
The nonparallel sides of a trapezoid.
Isosceles Trapezoid
Trapezoid with congruent legs. 

56. Trapezoids and Kites Base Angles of an Isosceles Trapezoid Theorem
If a trapezoid is isosceles, then each pair of base angles is congruent. D A B C 
A  B, C  D

57. Trapezoids and Kites Congruent Base Angles in a Trapezoid Theorem.
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. D A B C 
ABCD is an isosceles trapezoid.

58. Trapezoids and Kites Diagonals in an Isosceles Trapezoid Theorem
A trapezoid is isosceles if and only if its diagonals are congruent. D A B C 
ABCD is isosceles if and only if
AC  BD.

59. Trapezoids and Kites Midsegment of a trapezoid
The segment that connects the midpoints of a trapezoids legs.
midsegment

60. Trapezoids and Kites Midsegment Theorem for Trapezoids
The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
MN || AD, MN || BC,
MN = ½(AD + BC) B C D N M A 

61. Trapezoids and Kites Kite
A quadrilateral with two distinct pairs of consecutive congruent sides. Opposite sides are not congruent.

62. Diagonals of a Kite Theorem
If a quadrilateral is a kite, then its diagonals are perpendicular.

63. Opposite Angles in a Kite Theorem
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

64. Trapezoids and Kites Example 4 GHJK is a kite. Find HP. 5 √29 G H P J

65. Trapezoids and Kites Example 5 RSTU is a kite. Find mR, mS, and mT.
125° x° R S T U

66. 6.6 Special Quadrilaterals
Geometry 1 Unit 6
6.6 Special Quadrilaterals

67. Special Quadrilaterals

68. Special Quadrilaterals
Property
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Kite
1.Both pairs of opposite sides are congruent
2. Diagonals are congruent
3. Diagonals are perpendicular

69. Special Quadrilaterals
Property
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Kite
4. Diagonals bisect each other
5. Consecutive angles are supplementary
6. Both pairs of opposite angles are congruent

70. 6.7 Areas of Triangles and Quadrilaterals
Geometry 1 Unit 6
6.7 Areas of Triangles and Quadrilaterals

71. Area
Area is the number of square units in a figure.

72. Area-Rectangles Count the number of squares to find the area.
A shortcut is to find the length and multiply it by the width.

73. Area-Parallelograms Count the number of squares to find the area.
A shortcut is to find the length and multiply it by the width.

74. Rectangles and Parallelograms
Area = length times width
Area = base times height
A = bh b h

75. Area-Triangles
Have students cut out triangles from rectangles to see this property in action.

76. Area-Triangles
A = ½ base times height
A = ½ bh b h h h b b

77. Area-Trapezoids A = triangle 1 + triangle 2 A = ½ h(b1) + ½ h(b2)
A = ½ h(b1 + b2)

78. Area-Trapezoids A = ½ height times (base1 + base 2) A = ½ h(b1 + b2)

79. Area- Kites A = ½ (diagonal 1) times (diagonal 2) A = ½ (d1)(d2)
d1 and d2

80. Area-Rhombus A = ½ (diagonal 1) times (diagonal 2) A = ½ (d1)(d2)
d1 and d2

81. Example 1
Find the area of ΔRST. 3 4 T S R

82. Example 2
What is the base of a triangle that has an area of 48 and a height of 3?

83. Example 3
A rectangle has an area of 100 square meters and a height of 25 meters. Are all the rectangles with these dimensions congruent?

84. Example 4
Find the area of parallelogram RSTU. U R 6 3 S T

85. Example 5
What is the height of a parallelogram that has an area of 96 square feet and a base length of 8 feet?

86. Example 6 Find the area of trapezoid EFGH.

87. Example 7 Find the area of kite ABCD.

88. Example 8
Use the information given in the diagram to find the area of kite ABCD. A D C B 8 16

89. Example 9
The tray below is designed to save space on cafeteria tables. How much table area does the tray use?
16 in
8 in
3 in
10 in

90. Example 10
Find the area of rhombus EFGH
if EG = 10 and FH = 15.