Download Parallelograms Keystone Geometry 1 Parallelogram

Parallelograms
Keystone Geometry
1
Parallelogram
Definition: A quadrilateral with both pairs of opposite sides
parallel.
B
C
AB CD and BC AD
A
D
 The symbol for parallelogram is a smaller version of the figure.
 A parallelogram is named using all four vertices.
 You can start from any one vertex, but you must continue in a
clockwise or counterclockwise direction.
 For example, the figure above can be either
ABCD or
ADCB.
2
Properties of
Parallelogram
A
D
1.
B
P
C
Both pairs of opposite sides are congruent.
AB @ CD and BC @ AD
2.
Both pairs of opposite angles are congruent.
Ð A @ ÐC
3.
and ÐB @ ÐD
Diagonals bisect each other but are not congruent
P is the midpoint of AC and BD, so BP = PD
and AP = PC.
3
H
Examples
K
M
P
L
Draw
HKLP.
PL
KL
HK = _______
and HP = ________
.
P
m< K = m<______
.
P or K = 180°.
m< L + m<______
65° and
115° m< K = ____
If m< P = 65°, then m< H = ____,
115°
m< L =____.
Draw the diagonals with their point of intersection labeled M.
units .
If HM = 5, then ML = 5
____
(Alternate interior
14 units .
If KM = 7, then KP = ____
angles are congruent.)
units .
If HL = 15, then ML = 7.5
____
36° .
If m<HPK = 36°, then m<PKL = _____
4
Examples
• ABCD is a parallelogram. Find the values of x,
3z - 18
y, and z.
2x+46
y2-16
33
3x-10
A
8z-53
B
5
Example 2: In
GEOM, EA = 2x2 + 5x and AM = 4x + 2.
Find the value of x. What are EA and AM?
G
M
A
E
O
Example 3. Given
25, solve for x.
T
3
4
TRAC. If TC = 3x - 5 and RA =
R
K
1
2
C
A
If m∠CTR = 2y - 13 and m∠ACT = y + 1. Find y.
Example 4
In triangle ABC, M, N, and Q are midpoints.
MN is parallel to AC and MQ is parallel to BC.
B
mÐNMQ = 87 and AC=12, NC=5
and ÐA=27 .
M
N
Find :
MQ = _____
BC = _____
A
Q
C
ÐB = _____
ÐMQC = _____
ÐAMQ = _____
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