Properties 1. opposite sides II 2. opposite sides 3. opposite angles 4. diagonals bisect each other 5. consecutive angles supplementary Theorems (Ways to Prove) 1. If both pairs oppsite sides are ll … 2. If both pairs opposite sides … 3. If both pairs of opposite angles … 4. If diagonals bisect each other … 5. If one pair of sides are and ll … Parallelogram Parallelogram – quadrilateral with two pairs of parallel sides … then it’s a parallelogram
Transcript
Properties
1. opposite sides II
2. opposite sides
3. opposite angles
4. diagonals bisect each other
5. consecutive angles supplementary
Theorems (Ways to Prove)
1. If both pairs oppsite sides are ll …
2. If both pairs opposite sides …
3. If both pairs of opposite angles …
4. If diagonals bisect each other …
5. If one pair of sides are and ll …
Parallelogram
Parallelogram – quadrilateral with two pairs of parallel sides
… then it’s a parallelogram
Copy this one on the other Half!!
2. Explain how slope can be used to identify parallelograms in the coordinate plane.
3. Find the values of x and y that ensure each quadrilateral is a parallelogram.
a. b.
(2x + 8)o
120o
5y
If opposite sides have the same slope then it is a parallelogram.
y
y2
4x + 86x
6x = 4x + 8
2x = 8
X = 4
y2 = y
y2 – y = 0
y(y-1) = 0
y = 0 or y-1 = 0
y = 0 or y = 1
2x + 8 = 120
2x = 112
x = 56
5y + 120 = 180
5y = 60
y = 12
Properties
1. all properties of parallelogram.
2. four right angles
3. diagonals are .
Theorems (Ways to Prove)
1. If has 4 right angles, then it’s a rectangle.
2. If diagonals are , then it’s a rectangle.
Rectangle
Rectangle – quadrilateral with four right angles
Copy on the back of Rectangle Flap.
For 1-3 use quadrilateral MNOP is a rectangle. Find the value of x.
1. MO = 2x-8; NP = 23
2. CN = x2+1; CO = 3x+11
3. MO = 4x-13; PC = x+7
M N
OP
C
2x – 8 = 23
2x = 31
X = 15.5
x2 + 1 = 3x + 11
x2 - 3x - 10 = 0
(x-5)(x+2) = 0 x = 5 or x = -2
4x – 13 = 2(x + 7)
4x – 13 = 2x + 14
4x – 13 = 2x + 14
2x = 27
2x = 27
x = 13.5
Properties
1. all properties of a parallelogram.
2. all sides .
3. diagonals are ┴.
4. diagonal bisect opposite angles.
Theorems (Ways to Prove)
1. If it has 4 sides, then it’s a rhombus.
2. If diagonals are ┴, then it’s a rhombus.
3. If the diagonal bisects each pair of opposite angles, then it’s a rhombus.
Rhombus
Rhombus – a quadrilateral with four congruent sides
1. Use rhombus BCDE and the given information to find each missing value.
a. If m1 = 2x + 20 and m2= 5x – 4, find the value of x.
b. IF BD = 15, find BF.
c. If m3 = y2 + 26, find y.
B C
D
E
F
3
12
2x + 20 = 5x – 4
-3x = -24
x = 8
BF = ½ BD; BF = ½ (15); BF = 7.5
y2 + 26 = 90
y2 = 64
y = 8 or -8
Copy on the back of Rhombus
Flap.
Properties
All the properties of a parallelogram, rectangle, and rhombus
Ways to Prove:
1. If it is a rectangle and a rhombus, then it is a square.
Squares
45o45o
45o
45o
45o 45o
45o
45o
Notice you create 45-45-90 triangles
Remember: 1:1:√2
11
√2
Square – a quadrilateral with four right angles & four
congruent sides
Properties of SquaresOn the back, you will have one example:
First, if you know it’s a square, you can
mark what you know about a square:
Kite – a quadrilateral with two pairs of congruent consecutive sides.
Theorems: If a quadrilateral is a kite,
1. then diagonals are perpendicular
2. then one pair of opposite angles is congruent
Trapezoids
1. One pair of ll sides called bases.
2. The nonparallel sides are called legs.
Isosceles Trapezoids
1. The Legs are .
Theorems:
1. If isosc. Trap., then base angles are .
2. If base angles are , then it’s an isosc. Trap.
3. A trap. is isosc. If and only if the diagonals are .
4. The midsegment is one half the sum of the bases. M= ½ (base1 + base2)
Trapezoids
Legs Legs
Base angle Base angle
Base angleBase angle
base1
base2
Midsegment
1. Given the trapezoid EZOI with median AB, find the value of x.
E Z
OI
A B
4x-10
13
3x+18
AB = ½ (EZ + IO)
13 = ½ (4x - 10 + 3x + 18)
13 = ½ (7x + 8)
26 = 7x + 8
18 = 7x
18/7 = x
Copy on the back of Trapezoid Flap.
Properties of Trapezoids2. Given IEZO is an isosceles trapezoid, and
