Transcript To use relationships among sides and angles of

6-2 Vocabulary Support Choose the concept from the list above that best represents the item in each box.
Concept List 
congruent angles
diagonal
congruent segments is parallel to
opposite sides
parallelogram

1. A and B
2.
4. 𝑬𝑭 and 𝑯𝑮
5. H and F
7. 𝑾𝑿 = 𝒀𝒁
8.
consecutive angles
opposite angles
transversal
3.
EFGH
6. mH = mF
9.
6-2 Properties of Parallelograms
OBJECTIVES: To use relationships among sides and angles of parallelograms; & to use relationships among diagonals of parallelograms.
 Parallelograms have special properties regarding their sides, angles, and diagonals.
Parallelograms and Their Parts
Term Description
A parallelogram is a quadrilateral with both
pairs of opposite sides parallel.
You can abbreviate parallelogram with the symbol ▱.
In a quadrilateral, opposite sides do not share a vertex
and opposite angles do not share a side.
Angles of a polygon that share a side are consecutive angles.
In the diagram, ∠A and ∠B are consecutive angles
because they share side AB.
Diagram
6-2 Properties of Parallelograms
Theorem 6-3
If a quadrilateral is a parallelogram,
then its opposite sides are congruent.
K
E
Y
C
O
N
C
E
P
T
S
Theorem 6-4
If a quadrilateral is a parallelogram,
then its consecutive angles are supplementary.
Theorem 6-5
If a quadrilateral is a parallelogram,
then its opposite angles are congruent.
Theorem 6-6
If a quadrilateral is a parallelogram,
then its diagonals bisect each other.
Theorem 6-7
If three (or more) parallel lines cut off
congruent segments on one transversal,
then they cut off congruent segments
on every transversal.
6-2 Properties of Parallelograms
OBJECTIVES: To use relationships among sides and angles of parallelograms; & to use relationships among diagonals of parallelograms.
page 220
Using Consecutive Angles
Suppose you adjust the lamp in Problem 1 so that mS = 86. What is mR in
PQRS?
Answer: 94
Read the Got It and point out the consecutive angles, opposite sides, and
supplementary angles in the diagram. How does visual support help you
understand these terms? Which of these angles will help you solve the Got It?
Answer: The angles are consecutive and are therefore supplementary.
Using Properties of Parallelograms in a Proof
ABCD
Given
𝑨𝑲 ≅ 𝑴𝑲
Given
𝑩𝑨𝑫 ≅ 𝑩𝑪𝑫
Opposite s are ≅
in a parallelogram.
𝑩𝑨𝑫 ≅ 𝑪𝑴𝑫
Isosceles Triangle
Theorem
Type equation here.
𝑩𝑪𝑫 ≅ 𝑪𝑴𝑫
Substitution Property
of Congruence
6-2 Properties of Parallelograms
OBJECTIVES: To use relationships among sides and angles of parallelograms; & to use relationships among diagonals of parallelograms.
Using Algebra to Find Lengths
a. Find the values of x and y in
PQRS. What are PR and SQ?
Q𝑇 = 𝑆𝑇
 𝑦 =𝑥+1
𝑃𝑇 = 𝑇𝑅
 3𝑦 − 7 = 2𝑥
3 𝑥 + 1 − 7 = 2𝑥
3𝑥 + 3 − 7 = 2𝑥
3𝑥 − 4 = 2𝑥
𝒙=𝟒
𝑃𝑅 = 3𝑦 − 7 + 2𝑥
𝑃𝑅 = 3 𝟓 − 7 + 2(𝟒)
𝑷𝑹 = 𝟏𝟔
𝑺𝑸 = 𝟏𝟎
𝐲=𝟓
b. In part (a), does it matter which variable you solve for first? Explain.
No, system of equations do not depend on the method used for solving. Answers will always be the same.
Using Parallel Lines and Transversals
Use the figure below. If EF = FG = GH = 6 and AD = 15, what is CD?
Answer: 5
6-2 Properties of Parallelograms
OBJECTIVES: To use relationships among sides and angles of parallelograms; & to use relationships among diagonals of parallelograms.
1.
Find mA, mD, and AB.
Answers: mA = 53
mD = 127
AB = 7
2. A parallelogram has an angle whose measure is 4 times the measure of another angle of the parallelogram.
What are the measures of the angles of the parallelogram?
Answer: 36, 144, 36, 144
3. The director of a marching band uses a coordinate plane to plan the band’s shows. During the half-time show at a football
game, the four tuba players will form JKLM. The coordinates of three tuba players are shown. What are the coordinates of
the fourth tuba player?
Answer: 0, −3
6-2 Properties of Parallelograms
OBJECTIVES: To use relationships among sides and angles of parallelograms; & to use relationships among diagonals of parallelograms.
4. Vocabulary What is the difference between a quadrilateral and a parallelogram?
Both quadrilateral and parallelogram have four sides,
but if both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram.
5. Use Representations to Communicate Mathematical Ideas (1)(E)
Your classmate says that QV = 10 cm in the figure below. Use the figure to explain
why the statement may not be correct.
It is not given that 𝑃𝑄, 𝑅𝑆, and 𝑇𝑉 are parallel.
6. Justify Mathematical Arguments (1)(G)
Is it possible for a parallelogram to have three angles that each measure 75? Justify your response.
No; the 4th angle has to measure 75 too but it will contradict the Polygon Angle-Sum Theorem.
p. 260 Lesson 6-2 Properties of Parallelograms
20. Apply Mathematics (1)(A) A pantograph is an expandable device,
shown at the right. Pantographs are used in the television industry in
positioning lighting and other equipment. In the photo, points D, E, F, and G
are the vertices of a parallelogram. DEFG is one of many parallelograms
that change shape as the pantograph extends and retracts.
a. If DE = 2.5 ft, what is FG?
b. If mE = 129, what is mG?
c. What happens to mD as mE increases or decreases? Explain.
6-2 Properties of Parallelograms
Multiple Choice
1. One of the corners of a parallelogram-shaped coffee table has an angle measure of 53.
What is the angle measure of the opposite corner?
A. 37
2.
B. 53
C. 127
D. 307
H. 8
J. 4
What is the value of x in QRST?
F. 16
G. 12
3. Quadrilateral LMNO has vertices at 𝐿 0, 0 , 𝑀
for O makes LMNO a parallelogram?
A. 0, 1
B.
1
,1
2
1
,1
2
, and 𝑁 1, 1 . Which of the following coordinates
C. 0,
1
2
D. none of these
6-2 Properties of Parallelograms
Multiple Choice
4. What is m1 in this parallelogram?
F. 20
G. 60
H. 80
J. 100
C. 15
D. 2
5. What is m2 in this parallelogram?
A. 115
B. 50
Extended Response
6. Figure ABCD is a parallelogram. What are four geometric attributes you know because ABCD is a parallelogram?
 opposite sides are congruent
 𝐴𝐵 𝐷𝐶 and 𝐴𝐷 𝐵𝐶
 opposite sides are congruent
 𝐴𝐵 ≅ 𝐷𝐶 and 𝐴𝐷 ≅ 𝐵𝐶
 opposite angles are congruent
 A ≅ C and B ≅ D
 consecutive angles are supplementary
 mA + mB = 180, mB + mC = 180,
mC + mD = 180, and mA + mD = 180