# Theorem 5 - Xavier High School

```5-3 Theorems involving
parallel lines
Theorems involving parallel lines
Theorem 5 – 8
If two lines are parallel, then all points on one line are equidistant from the
other line
l
A
B
C
D
m
Since line l is parallel to line m; AC = BD
Theorems involving parallel lines
Theorems 5 – 9
If three parallel lines cut off congruent segments on one
transversal, then they cut off congruent segments on every
transversal
A
B
3
X
5 Y
C 4
Given AX
BY
CZ, and AB = BC we can say that XY = YZ
6
Z
Theorems involving parallel lines
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 a third side
A
M
B
D
N
C
If M is the midpoint of AB and MN BC. We can draw a third line from A called
D, so that AD, MN, and BC are all parallel. Now we have three parallel
lines cut by a transversal. We can use theorem 5-9 to say that N is the
midpoint of AC
Theorems involving parallel lines
Theorem 5 – 11
The segment that joins the midpoints of two sides of a triangle
(1) is parallel to the third side;
(2) is half as long as the third side
A
M
B
N
C
If N is the midpoint of AB and N is the midpoint of AC we can prove that MN is
parallel to BC because there is exactly one line through M parallel to BC.
By Theorem 5 – 10 that line passes through N and the midpoint AC. So MN
is parallel BC
Example
P, Q, and R are midpoints of the sides of
DEF.
12
P
D
a)
b)
E
What kind of figure is DPQR
What is the Perimeter of DPQR
R
Q
8
10
F
Solution
P, Q, and R are midpoints of the sides of
DEF.
12
D
a)
b)
E
P
What kind of figure is DPQR
What is the Perimeter of DPQR
Q
R
8
10
F
a)
Since RQ DE and PQ
DF, Quad DPQR is a parallelogram
b) RQ = &frac12; DE = DP = 6 and PQ = &frac12; DF = DR = 4.
DPQR is 6 + 4 + 6 + 4 or 20
So the perimeter of
```