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Lesson 5-3
Theorems Involving Parallel Lines
(page 177)
Essential Question
How can the properties of quadrilaterals
be used to solve real life problems?
Properties of Parallelograms
• Parallelogram has both pairs of opposite sides parallel.
• Parallelogram has both pairs of opposite sides congruent.
• Parallelogram has both pairs of opposite angles congruent.
• Parallelogram’s diagonals bisect each other.
• The consecutive angles of a parallelogram are
supplementary.
5 Ways to Prove that a
Quadrilateral is a Parallelogram
 Show that both pairs of opposite sides are parallel.
 Show that both pairs of opposite sides are congruent.
 Show that one pair of opposite sides are both ≅ & || .
 Show that both pairs of opposite angles are congruent.
 Show that the diagonals bisect each other.
Theorem 5-8
If two lines are parallel, then all points on one line
are equidistant from the other line.
Theorem 5-9
If three parallel lines cut off congruent segments on one transversal,
then they cut off
congruent
segments on every transversal.
A
B
C
Proof:
➤
➤
➤
X
Y
Z
For an outline of this proof see page 177.
Note that parallelograms and congruent triangles are used in the proof.
We will do an informal proof using the parallel lines on notebook paper.
Theorem 5-10
A line that contains the midpoint of one side of a triangle and is
parallel to another side passes through the midpoint of the third side.
A
Given: M is the midpoint of
AB
M
Prove: N is the midpoint of
AC
B
➤
➤
D
➤
●
N
C
Theorem 5-11
The segment that joins the
(1) is
(2) is
midpoints of two sides of a triangle:
parallel to the third side;
half as long as the third side.
A
Given: M is the midpoint of AB
N is the midpoint of AC
B
M
N
C
Theorem 5-11
The segment that joins the
(1) is
(2) is
midpoints of two sides of a triangle:
parallel to the third side;
half as long as the third side.
A
Given: M is the midpoint of AB
N is the midpoint of AC
B
M
➤
➤
N
C
The segment that joins the midpoints of two sides of a triangle:
(1) is parallel to the third side;
A
Given: M is the midpoint of AB
N is the midpoint of AC
Prove: (1)
B
M
➤
N
➤
Proof: To prove this part of the theorem, use Theorem 5- 10
reasoning indirectly to prove this part of the theorem.
C
Theorem 5-11
The segment that joins the
(1) is
(2) is
midpoints of two sides of a triangle:
parallel to the third side;
half as long as the third side.
A
M
B
N
C
The segment that joins the
(2) is
midpoints of two sides of a triangle:
half as long as the third side.
A
M
B
N
●
L
C
Example #1:
If NT = x + 6 and NZ = 3 x - 8, find the value of x, NT, TZ, and NZ.
N
(x+6)(3x-8)
T
(x+6)
Z
M
S
NT + TZ = NZ
x+6+x+6 = 3x-8
2x+12 = 3x-8
20 = x
20
26
NT = ______
26
TZ = ______
52
NZ = ______
x = ______
Y
NT = TZ = x+6 = 20+6 = 26
NZ = 3x-8 = 3(20)-8 = 60-8 = 52
Example #2
Name the points shown that must be midpoints of the sides of the large triangle.
X
A&C
A
Y
B
6
C
6
Z
Example #3:
If ST = 4 x + 4 and AB = x + 40, find x, ST, and AB.
R
38
78
AB = ______
ST = 156
______
x = ______
A
S
B
T
Assignment
Written Exercises on pages 180 & 181
RECOMMENDED: 1, 3, 17, 19
REQUIRED: 5 to 15 odd numbers, 20
Worksheet on Lessons 5-1 to 5-3: Parallelograms
Prepare for Quiz on Lessons 5-1 to 5-3
How can the properties of quadrilaterals
be used to solve real life problems?