Transversal:
Name the obvious transversal(s):
c
1.
a
a
2.
b
3.
a
c
b
c
b
When 2 coplanar lines are cut by a transversal, 8 angles are formed:
INTERIOR ’s:
EXTERIOR ’s:
Some of these angles have a relationship that we have previously studied.
LINEAR PAIRS:
VERTICAL ’s:
c
b
6
7
a
5
8
1 2
3 4
1
Types of Angles
Alternate Interior Angles:
Alternate Exterior Angles:
Same-side Interior Angles (consecutive angles):
Corresponding Angles:
Use the given line as a transversal:
1. Name alt. int ’s using line x:
2
2. Name s.s. int. ’s using line y:
1
3
4
3. Name corr. ’s using line z:
6
4. Name alt. ext. ’s using line y:
8 7
5. Name alt. int. ’s using line z:
5
10
9
y
x
z
6. Name s.s. int ’s using line z:
12
11
Examples – Find the measures of the angles (or value of the variables(s)).
1.
5.
3.
2.
6.
7.
4.
8.
2
9.
10.
11.
3
Is it possible to prove the lines are parallel or not parallel? If so, state the postulate or theorem
you would use. If not, state cannot be determined.
1.
2.
3.
92°
l
k
88°
4.
5.
6.
l
k
7. A
105°
E
8.
B
I
C
D
9.
k
122°
75°
H
55°
l
F
58°
55°
m
G
Find the value of x so that n || m. State the theorem or postulate that justifies your solution.
5x
10.
n
11.
m
5x+23
n
7x+13
m
12.
5x-18
n
m
8x-5
3x+48
x=
x=
x=
_____________________
_____________________
_____________________
Can you prove that lines p and q are parallel? If so, state the theorem or postulate that you
would use.
p
13.
p
14.
q
q
_____________________
p
15.
q
_____________________
_____________________
4
3.3 Parallel Proofs
t
l
1
2
m
3
4
5
1. Given: l // m; 1  4
Prove: s // t
1. l // m ; 1  4
1. _______________________________________
2. 3  1
2. _______________________________________
3. 3  4
3. _______________________________________
4. s // t
4. _______________________________________
2. Given: l // m ; 2  5
Prove: s // t
1. l // m ; 2  5
1. ________________________________________
2. 2  3
2. ________________________________________
3. 3  5
3. ________________________________________
4. s // t
4. ________________________________________
3. Given: l // m; s // t
Prove: 2  4
1. l // m ; s // t
1. ________________________________________
2. 2   3
3  4
3. 2  4
2. ________________________________________
3. ________________________________________
4. Given: l // m; s // t
Prove: 1  5
1. l // m; s // t
1. _________________________________________
2. 1  3
3  5
3. 1  5
2. _________________________________________
3. _________________________________________
5
5. Given: 3 is supplementary to 5.
Prove:
BD // FE
1. 3 is supplementary to 5
1. _________________________________________
2. m3 + m5 = 180
2. _________________________________________
3. 3  4
3. _________________________________________
4. m3 = m4
4. _________________________________________
5. m4 + m5 = 180
5. _________________________________________
6. 4 is supplementary to 5
6. _________________________________________
7.
7. _________________________________________
BD // FE
B
A
6. Given: 2  5;
BE bisects  CBD.
Prove: AC // DE
1
D
1. 2  5;
BE bisects  CBD.
2.
3. 3  5
3.
4.
4.
AC // DE
8. Given: l // m ; s // t
Prove: 1  5
9. Given: l // m; 1  4
Prove: s // t
C
3
4
5
E
1.
2. 3  2
7. Given: l // m ; s // t
Prove: 2  4
2
s
Diagram for # 7 - 10
l
t
1
2
m
3
5
4
10. Given: l // m; 2  5
Prove: s // t
6
11. Given: BC // EF ; BA // ED
Prove: B  E
A
D
C
P
B
E
12. Given: AB //DE
Prove: mACD = mBAC + mCDE
F
A
B

C

E
D
13. Given: g // h; g // j
Prove: 2  3
g
1
2
h
3
14. Given: AB //CD ; BC //DE
Prove: B  D
j
C
A
E
B
D
15. Given: a // c; 1  2
Prove: b // c
2
1
a
16. Given: C is a supplement of D
Prove: A is a supplement of B
b
B
A
c
C
D
7
3.4 Perpendicular Lines
Perpendicular Bisector is a line perpendicular to a segment at the segment's midpoint.
Distance from a point to a line the length of the perpendicular segment from the point to the line.
1- 6 Use the given diagram on the right, in which AM = MB.
 C
1. Name a pair of  rays.
2. ____ is the  bisector of ____.
3. Name a linear pair of angles which are .
4. If t in X is  to
5. If

A
M

B
AB at M, what can you say about t and CM ? Why?
MR in X is a  bisector of AB , then R is on CM . Why?
6. If p contains M and is  to the plane determined by
CM and AB , then p ___ CM and p ___ AB . Why?
7. In a plane , how many lines can be  to a given line at a given point?
8. Would your answer be different if the words “in a plane” were omitted from the question?
Homework on Perpendicular Lines.
True or False. If false, give a counterexample.
_____1. If
PQ  PR , then QPR is a right angle.
______2. If
AB  CD , then ABC is a right angle.
_____3. If 2 lines intersect to form a right angle,
then the lines are .
______4. There is exactly one line  to a given at a
given point on the line.
_____5. If 2 angles are a linear pair, then each
is a right .
______6. A given segment has exactly one 
bisector.
_____7. If M is the midpoint of AB and if AB is  to
plane X at M, there is exactly one line in X which
______8. If 2 adjacent angles are , then each is a
right angle.
is a  bisector of
AB .
8
In 9 – 13 refer to the diagram below and the given info. : mCAB = 90; CDA  BDA;
EA  AB ; mECB = 90
C
**Mark the diagram with the given information**
X
9. What pairs of lines are ?
D
10. ____ is a  bisector of ____. Why?
E
A
11. If
FC in X is a  bisector of EB , then F is on AC . Why?
12. If
t is a line in the plane of the diagram, and t  BC at D, how are t and AD related? Why?
13. If G is on
B
CE and EGA  CGA, how are AG and EC related? Why?
14. If
l  m and m  n, is l  n? Explain.
15. If
m  n, is n  m ? Explain.
For Exercises 3 and 4, name the shortest segment from the point to the line and
write an inequality for x.
3.
4.
________________________________________
________________________________________
9
3.5-3.6 Slopes of Lines
10
Write the equation of each line in the given form.
1. the horizontal line through (3, 7) in
2. the line with slope 
point-slope form
8
through (1, 5) in
5
point-slope form
 1 7
3. the line through   ,   and (2, 14) in
 2 2
slope-intercept form
4. the line with x-intercept 2 and y-intercept
1 in slope-intercept form
Show All Work! Solve.
1. x2 – 81 = 0
2. x2 – 7x + 10 = 0
3. x2 + 10x + 25 = 0
4. 10x – 24 = x2
5. 6x2 – 5x + 1 = 0
6. x2 – 3x = 0
7. x2 – x – 12 = 0
8. x2 – 2x = 3
9. x2 + 49 = 14x
10. 2x2 + 7x = -6
11. 6x2 – x = 12
12. 10x2 + 3x = 18
In the problems, l // m. Find x.
t
1
2
4 3
l
5
6
8
m
7
13. m4 = x2 + 72; m5 = -16x + 171
14. m3 = x2 – 2x ; m6 = 3x + 108
15. m3 = x2 + 2x ; m7 = -x + 70
16. m4 = 2x2 + 10; m6 = 11x – 5
1. Name the slope and y-intercept.
x
a) y   1
b) 3x + 2y = 6
4
c) y – 4 =
2
(x  2)
3
2. Write an equation of the line in Standard Form.
a) m = -2 , b = 5
b) m = -
2
and the line contains (9,3)
5
11
c) the line contains (2,4) and (-3, 1)
d) M(-2, 1) , N(2,4). Write the equation of the line // to MN with y-intercept of –4.
e) Write an equation of the line through K(1,-2)  to MN .
f) A(1,-5) , B(-3, 5). Write an equation of the  bisector of AB .
1. The vertices of a ∆ are A(-1,2), B(-1, 8) and C(3,5). Classify the ∆ by its sides.
2. The vertices of ∆JKL are J(-5,0), K(5,8) and L(4,-1). Is ∆JKL equilateral?
3. Find the perimeter of ∆GHI with vertices: G(8,5), H(-1,-4), and I(-4,0). Exact answer!
4. The vertices of ∆ABC are A(-2,-1), B(-6,7) and C(4,2). Prove that ∆ABC is a right ∆.
5. Find the slope of a line parallel to AB , given A(-5,4) and B(3,6).
6. Given: A(-4,-3), B(-2,-9), C(-4,1) and D(-7,0) Are AB and CD //, , or neither?
7. Write each equation in Standard Form:
a) y = 3x + 6
b) y  3 
5
( x  3)
3
c) -y =
1
x 5
2
8. Write an equation of the line through the point (-4,5) // to y = 2x + 6 .
9. Write an equation of the line through the point (3,-1)  to x – 3y = -2
10. Write the equation of the line through A(4,1) // to the line through B(-2,3) and C(-4,-3).
11. Write the equation of the  bisector of AB , given A(5,2) and B(-1,4).
12. Find the point of intersection of 3x+y = 5 and the line containing (8,1) with a slope of
1
.
3
12