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Chapter 3 Notes
Mrs. Myers – Geometry
Name ______________________________
Period ______
3.1 Lines and Angles

Parallel Lines: when two lines are coplanar and do not intersect.

Skew Lines: lines that do not intersect and are not coplanar.

Parallel Planes: two planes that do not intersect.
Ex. 1 Identify relationships in space.
A)
to AB and point D.
B)  to AB and point D.
C) skew to AB and point D.
D) plane that is
to plane ABEF and point D.
* Postulate 13: Parallel Postulate = If there is a line and a point not on the line, there is
exactly one line through the point parallel to the given line.
* Postulate 14: Perpendicular Postulate = If there is a line and a point not on the line,
there is exactly one line through the point perpendicular to the given line.

Transversal: is a line that intersects two or more coplanar lines at different
points (line t).

Corresponding Angles: two angles that occupy corresponding positions.

Alternate Exterior Angles: two angles that lie outside the two lines on opposite
sides of the transversal.

Alternate Interior Angles: two angles that lie between the two lines on opposite
sides of the transversal.

Consecutive Interior Angles: two angles that lie between the two lines on the
same side of the transversal.
Ex. 2 Identify angle relationships
A) Corresponding
B) Alternate Exterior
C) Alternate Interior
D) Consecutive Interior
3.2 Proof and Perpendicular Lines
* Theorem 3.1: if two lines intersect to form a linear pair of congruent angles, then the
lines are perpendicular.
* Theorem 3.2: if two sides of two adjacent acute angles are perpendicular, then the
angles are complementary.
* Theorem 3.3: if two lines are perpendicular, then they intersect to form four right
angles.
Ex. 1 Proof of theorem 3.1
Given:
1
2
1 and
2 are a linear pair
Prove: G  H
1.
1 and
Statements
2 are a linear pair
1. Given
2.
1 and
2 are sup plementary
2.
3. m 1  m 2  180
4.
1
2
Reasons
3.
4. Given
5. m 1  m 2
5.
6. m 1  m 1  180
6.
7. 2  m 1  180
7. Simplify / combine like terms /
distributive property of equality
8. m 1  90
8.
9.
9.
1 is a right angle
10. G  H
10.
Ex. 2
Given: BA  BC
Prove:
1. BA  BC
2.
1 and
2 are complementary
Statements
Reasons
1. Given
ABC is a right angle
2.
3.
3. Definition of right angles
4. m ABC  m 1  m 2
4.
5. 90  m 1  m 2
5. Substitution property of equality
6.
6.
1 and
2 are complementary
3.3 Parallel Lines and Transversals
* Postulate 15: Corresponding Angles Postulate = If two parallel lines are cut by a
transversal, then the pairs of corresponding angles are congruent.
If , then
1
2
* Theorem 3.4: Alternate Interior Angles Theorem = If two parallel lines are cut by a
transversal, then the pairs of alternate interior angles are congruent.
If
, then
3
4
* Theorem 3.5: Consecutive Interior Angles Theorem = If two parallel lines are cut by
a transversal, then the pairs of consecutive interior angles are supplementary.
If
, then m 5  m 6  180
* Theorem 3.6: Alternate Exterior Angles Theorem = If two parallel lines are cut by a
transversal, then the pairs of alternate exterior angles are congruent.
If , then
7
8
* Theorem 3.7: Perpendicular Transversal = If a transversal is perpendicular to one of
two parallel lines, then it is perpendicular to the other.
If h k and j  h , then j  k
Ex. 1 Given m 5  65 . Find the other measurements of the angles and state why.
A) m 6  _____ because of __________________________________________
B) m 7  _____ because of __________________________________________
C) m 8  _____ because of __________________________________________
D) m 9  _____ because of __________________________________________
Ex. 2 Proof of the Alternate Interior Angles Theorem
Given: P Q
Prove:
1
2
Statements
Reasons
1. Given
1. P Q
2.
1
3
2.
3.
2
3
3.
4.
1
2
4.
Ex. 3
Given: P Q
Prove: m 1  m 2  180
Statements
Reasons
1. Given
1. P Q
2.
1
3.
2&
4 are a linear pair
3. Definition of linear pair
4.
2&
4 are supplementary
4. Linear pair postulate
5.
1&
2 are supplementary
5.
2.
4
6. m 1  m 2  180
6.
Ex. 4
Given: diagram
Prove: x  40
Statements
1. m 4  x  15  180
Reasons
1. Linear pair postulate
2. m 4  125
2.
3. 125  x  15  180
3. Substitution property of equality
4. x  140  180
4. Simplify
5. x  40
5.
3.4 Proving Lines are Parallel

Postulate 16: Corresponding Angles Converse = If corresponding angles are
congruent, then the lines are parallel.
If

k
Theorem 3.8: Alternate Interior Angles Converse = If alternate interior angles
are congruent, then the lines are parallel.
If

' s are  , then j
1
3, then j
k
Theorem 3.9: Consecutive Interior Angles Converse = If consecutive interior
angles add up to 180 (supplementary), then the lines are parallel.
If m 1  m 2  180 , then j

k
Theorem 3.8: Alternate Exterior Angles Converse = If alternate exterior angles
are congruent, then the lines are parallel.
If
4
5, then j
k
Ex. 1
Given: m  p and m  q
Prove: p q
p
1
2
q
Statements
1. m  p and m  q
1. Given
2.
1 is a right angle
2.
3.
2 is a right angle
3. Definition of perpendicular lines
4.
1
4.
5. p
2
Reasons
5.
q
Ex. 2 Find the value of x that makes m n .
m
n
2x + 1
3x - 5
3.5 Using Properties of Parallel Lines
* Theorem 3.11: If two lines are parallel to the same line, then they are parallel to each
other.
If p
q and q
r , then p
r
* Theorem 3.12: In a plane, if two lines are perpendicular to the same line, then they are
parallel to each other.
If m  p and n  p, then m n
Ex.1 Proving theorem 3.11.
Given: m n, n k
Prove: m k
Statements
2.
1
Reasons
1. Given
1. m n
2
2.
3. n k
3. Given
4.
4. Corresponding angles postulate
5.
1
6. m k
3
5.
6.
Ex. 2 Another way to prove theorem 3.11.
Given: r s, s t
Prove: r t
r
1
2
s
t
3
Statements
1. r
2.
1
Reasons
1. Given
s
2
2.
3. s t
3. Given
4.
4.
5.
1
6. r t
3
5. Transitive property of 
6.
3.6 Parallel Lines in the Coordinate Plane

Slope =
rise
run
* m
y2  y1
x2  x1
Ex. 1 Find the slope of a line that passes through the points  3,0 and  4,7  .
* Postulate 17: Slope of Parallel Lines = In a coordinate plane, two nonvertical lines
are parallel if and only if they have the same slope. Any two vertical liens are parallel.
Ex. 2 If AB has two coordinates 1,1 and 3,5 and CD has two coordinates
 6, 4 and  7,6 , are the two lines parallel?

Slope-Intercept Form:
o
y  mx  b
where
m  slope
b  y  int ercept
Ex. 3 Write an equation of the line through the point  4,9  that has a slope of -2.
2
5
through the point  5, 0  . Write the equation of k2 .
Ex. 4 Line k1 has the equation y  x  3. Line k2 is parallel to line k1 and passes
3.7 Perpendicular Lines in the Coordinate Planes
* Postulate 18: Slope of Perpendicular Lines = In a coordinate plane, two nonvertical
lines are perpendicular if and only if the product of their slopes is -1 (opposite
reciprocals). Note: vertical and horizontal lines are perpendicular to each other.
Ex. 1 Is P  Q ?
Line P passes through points  1,3 and 1, 1 .
Line Q passes through points  2,0 and  2, 2 .
Ex. 2 Is PQ  RS ?
Point P:
Point Q:
Point R:
Point S:
 3,3
 5, 2 
 1,  4
 0,1
Ex. 3 Is J  N ?
2
Line J: y   x  1
3
3
Line N: y   x  1
2
Ex. 4 Is B  K B?
Line B: 2 x  7 y   4
Line K: 7 x  2 y  10
Ex. 5 Line G has the equation y  3x  2 . Find the equation of line H that passes
through  3, 4 and is perpendicular to G.