Transcript Slide 1

6.2 Parallelograms
Objectives
• Recognize and apply properties of
the sides and angles of
parallelograms.
• Recognize and apply properties of
the diagonals of parallelograms.
Key Vocabulary
• Parallelogram
Theorems
• 6.2 – 6.5 Properties of
Parallelograms
Introduction
• Any polygon with four sides is a
quadrilateral. However, some
quadrilaterals have special
properties. These special
quadrilaterals are given their own
names.
• Opposite sides of a quadrilateral do
not share a vertex. Opposite angles
do not share a side.
Parallelograms
What makes a polygon a
parallelogram?
Parallelogram
• A parallelogram is a quadrilateral with both
pairs of opposite sides parallel.
• When you mark diagrams of quadrilaterals,
use matching arrowheads to indicate which
sides are parallel. For example, in the
diagram to the right, PQ║RS and QR║SP.
The symbol
PQRS is read
“parallelogram PQRS.”
PROPERTIES OF
PARALLELOGRAMS
Properties of Parallelograms
Q
• Theorem 6.2 - If
a quadrilateral is
a parallelogram,
then its opposite
sides are
congruent.
►PQ≅RS and
SP≅QR
P
R
S
Properties of Parallelograms
Q
R
• Theorem 6.3 - If
a quadrilateral is
a parallelogram,
then its opposite
angles are
congruent.
P ≅ R and
Q ≅ S
P
S
Properties of Parallelograms
Q
• Theorem 6.4 - If a
quadrilateral is a
parallelogram, then
its consecutive
angles are
supplementary (add
up to 180°).
mP +mQ = 180°,
mQ +mR = 180°,
mR + mS = 180°,
mS + mP = 180°
P
R
S
Summary Parallelogram
Theorems
• Theorem 6.2 – Opposite sides of
• Theorem 6.3 – Opposite s in
• Theorem 6.4 – Consecutive s in
supplementary.
are ≅.
are ≅.
are
Example 1
FGHJ is a parallelogram. Find JH and FJ.
SOLUTION
JH = FG
Opposite sides of a
=5
Substitute 5 for FG.
FJ = GH
Opposite sides of a
=3
ANSWER
are congruent.
are congruent.
Substitute 3 for GH.
In
FGHJ, JH = 5 and FJ = 3.
Your Turn:
1.
ABCD is a parallelogram.
Find AB and AD.
ANSWER
AB = 9; AD = 8
Example 2
PQRS is a parallelogram. Find
the missing angle measures.
SOLUTION
1. By Theorem 6.3, the opposite angles of a parallelogram are congruent,
so mR = mP = 70°.
2. By Theorem 6.4, the consecutive angles of a parallelogram
are supplementary.
mQ + mP = 180°
Consecutive angles of a
supplementary.
mQ + 70° = 180°
Substitute 70° for mP.
mQ = 110°
are
Subtract 70° from each side.
Example 2
3. By Theorem 6.3, the opposite angles of a parallelogram are congruent,
so mS = mQ = 110°.
ANSWER
The measure of R is 70°, the measure of Q is 110°, and
the measure of S is 110°.
Your Turn:
ABCD is a parallelogram. Find the missing angle measures.
2.
ANSWER
mB = 120°; mC = 60°; mD = 120°
3.
ANSWER
mA = 75°; mB = 105°; mC = 75°
Example 3A
A. CONSTRUCTION In
suppose
mB = 32, CD = 80 inches, BC = 15 inches.
Find AD.
Example 3A
AD = BC
= 15
Answer: AD = 15 inches
Opposite sides of a
Substitution
are .
Example 3B
B. CONSTRUCTION In
suppose
mB = 32, CD = 80 inches, BC = 15 inches.
Find mC.
Example 3B
mC + mB = 180
mC + 32 = 180
mC = 148
Answer: mC = 148
Cons. s in a are
supplementary.
Substitution
Subtract 32 from each
side.
Example 3C
C. CONSTRUCTION In
suppose
mB = 32, CD = 80 inches, BC = 15 inches.
Find mD.
Example 3C
mD = mB
= 32
Answer: mD = 32
Opp. s of a
Substitution
are .
Your Turn
A. ABCD is a parallelogram. Find AB.
A. 10
B. 20
C. 30
D. 50
Your Turn
B. ABCD is a parallelogram. Find mC.
A. 36
B. 54
C. 144
D. 154
Your Turn
C. ABCD is a parallelogram. Find mD.
A. 36
B. 54
C. 144
D. 154
Diagonals of Parallelograms
• The diagonals of a parallelogram
have special properties as well.
• Next theorems using diagonals of
parallelograms.
Theorem 6.5
If a quadrilateral is a
parallelogram,
then its diagonals
bisect each other.
Point M is the midpoint
of both diagonals.
More on Diagonals
If a quadrilateral is a
parallelogram, then
each diagonal
separates the
parallelogram into
two congruent
triangles.
A
D
B
C
If ABCD is a parallelogram, then
∆ABD≅∆CDB.
More on Diagonals
• A parallelogram with
two diagonals
divides the figure
into pairs of
congruent triangles.
• If ABCD is a
parallelogram, then
∆AZD≅∆BZC and
∆AZB≅∆DZC.
A
B
Z
D
C
Example 4
TUVW is a parallelogram.
Find TX.
SOLUTION
TX = XV
=3
Diagonals of a
bisect each other.
Substitute 3 for XV.
Example 5A
A. If WXYZ is a parallelogram, find the value of r.
Opposite sides of a
parallelogram are .
Definition of congruence
Substitution
Divide each side by 4.
Answer: r = 4.5
Example 5B
B. If WXYZ is a parallelogram, find the value of s.
8s = 7s + 3
s=3
Answer: s = 3
Diagonals of a
each other.
bisect
Subtract 7s from each side.
Example 5C
C. If WXYZ is a parallelogram, find the value of t.
ΔWXY  ΔYZW
Diagonal separates a
parallelogram into
2  triangles.
YWX  WYZ
CPCTC
mYWX = mWYZ
Definition of congruence
Example 2C
2t = 18
t =9
Answer: t = 9
Substitution
Divide each side by 2.
Your Turn:
A. If ABCD is a parallelogram, find the value of x.
A. 2
B. 3
C. 5
D. 7
Your Turn:
B. If ABCD is a parallelogram, find the value of p.
A. 4
B. 8
C. 10
D. 11
Your Turn:
C. If ABCD is a parallelogram, find the value of k.
A. 4
B. 5
C. 6
D. 7
Assignment
• Pg. 313 - 315 #1 – 49 odd