today > 8.2 properties of parallelograms by definition, a parallelogram is a quadrilateral with 2...
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TODAY> 8.2 Properties of Parallelograms
By definition, a parallelogram is a quadrilateral with 2 pairs of parallel sides
Given: GEOM is a parallelogram.
Prove:
(i.e. opposite sides are )
GE OM GM OE G E
OM
Given: GEOM is a parallelogram. Prove:a) G and E are supplementary. E and O are supplementary. O and M are supplementary. M and G are supplementary. (i.e. consecutive angles are supplementary)
b) G O, M E (i.e. opposite angles are congruent)
G E
OM
Given: GEOM is a parallelogram.
Prove: Diagonals bisect each other.
G E
OM
T
8.2 //ogram Properties
2 pairs of opposite sides are // (by defn.) 2 pairs of consecutive interior s are
supplementary 2 pairs of opposite s are 2 pairs of opposite sides are The diagonals bisect each other.
Exercises: p. 512 #8, 11, 15, 23 – 28, 33, 36
TODAY> 8.3 Proving Parallelograms
Aside from using the definition of a parallelogram (opposite sides are parallel), there are five (5) other ways to prove that a quadrilateral is a parallelogram.
Given:
Quadrilateral GEOM
G and E are supplementary.
E and O are supplementary.
O and M are supplementary.
M and G are supplementary.
Prove: GEOM is a parallelogram.
G E
OM
Given:
Quadrilateral GEOM
M E and G O
Prove:
GEOM is a parallelogram.
G E
OM
b
ba
a
Given:
Quadrilateral GEOM
Diagonals bisect each other at T.
Prove:
GEOM is a parallelogram.G E
OM
T
Given:
Quadrilateral GEOM
Prove:
GEOM is a parallelogram.
GE OM GM OE G E
OM
Given:
Quadrilateral GEOM
Prove:
GEOM is a parallelogram.
GM EO G E
OM
GM EO
A quadrilateral is a parallelogram if:
2 pairs of opposite sides are // (by defn.) 2 pairs of consecutive interior s are
supplementary 2 pairs of opposite s are 2 pairs of opposite sides are The diagonals bisect each other. One pair of opposite sides are // and .
8.3 Proving Parallelograms
1. Given: ABCD is a parallelogram & .
Prove: AECF is a parallelogram.
Warm-up: p. 521 #15 – 18 BF DE
8.3 Proving Parallelograms
3. Given: ABCD is a parallelogram.
E and F are midpoints.
Prove: EFCD is a parallelogram.
A B
CD
E F
8.3 Proving Parallelograms
4. Given: JOHN is a parallelogram.
Prove: JBHD is a parallelogram. JD ON HB ON
J O
HN
BD
8.3 Proving Parallelograms
TODAY> 8.4 Special Parallelograms
Rhombus Properties
The diagonals are bisectors of each other. The diagonals bisect the angles of the rhombus.
Remember your P.T. & Special Right s.
Rectangle Properties
The measure of each of a rectangle is 90o. The diagonals of a rectangle are and bisect
each other.
How many Isosceles
s are there?
Square Properties
The diagonals of a square are , and bisect each other.
Exercises: p. 531
How many isosceles RIGHT s are there?
A-S-N (True)
1. The diagonals of a parallelogram are congruent.
2. The consecutive angles of a rectangle are congruent and supplementary.
3. The diagonals of a rectangle bisect each other.
4. The diagonals of a rectangle bisect the angles.
5. The diagonals of a square are perpendicular bisectors of each other.
6. The diagonals of a square divides it into 4 isosceles right triangles.
7. Opposite angles in a parallelogram are congruent.
8. Consecutive angles in a parallelogram are congruent.
SUMMARYParallelogram Rhombus Rectangle Square
Opp sides are //
Opp sides are
Opp s are
Diagonals bisect each other
Diagonals are
Diagonals are
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y Y
Y Y
Proving
1. Given: MPQS is a rhombus.
G, H, I and K are midpoints.
Prove: GHIK is a rectangle.S
BG
K Q
M P
I
H