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Geometry Unit 3 Parallel and Perpendicular Lines Lesson Notes
3 1 Lines and Angles
Objectives:
I will be able to identify relationships between figures in space.
I will be able to identify angles formed by two lines and a transversal.
Do Now
Solve for each variable. Explain your reasoning. 4b + 43 = 6b + 17 b = 13 7a + 8 = 8a
–
3 a = 11
Parallel lines
are coplanar lines that do not intersect. The symbol
∥
means “is parallel to.”
⃡ ∥ ⃡
⃡ ∥ ⃡
Skew lines
are noncoplanar; they are not parallel and do not intersect.
⃡
and
⃡
are skew.
Parallel Planes
are planes that do not intersect.
Plane
ABCD
∥
plane
EFGH
Consider the figure below.
a. Which three segments are parallel to
̅
?
̅
,
̅
,
̅
b. Which four segments are skew to
̅
?
̅
,
̅
,
̅
,
̅
c. What are two pairs of parallel planes? Name three pairs.
VYXW
∥
RUTS, WXTS
∥
VYUR, VWRS
∥
YXTU
d. What are two segments parallel to plan
RUYV
?
̅
,
̅
A
transversal
is a line that intersects two or more coplanar lines at distinct points. Angles
3
,
4
,
5
, and
6
lie between
l
and
m
. They are
interior
angles. Angles
1
,
2
,
7
, and
8
lie outside of
l
and
m
. They are
exterior
angles.
Transversal
: a line that intersects two or more coplanar lines at different points.
The angles formed by two lines and a transversal are given special names.
Corresponding Angle:
two a
ngles that
occupy corresponding positions.
∠
1
and
∠
5
are corresponding angles.
Alternate Exterior Angle:
two angles that lie outside the two lines on opposite sides of the transversal.
∠
2
and
∠
8
are alternate exterior angles.
List all pairs of angles that fit the description.
Alternate Interior Angles:
two angles that lie between the two lines on opposite sides of the transversal.
∠
3
and
∠
5
are alternate interior angles.
corresponding alternate exterior Alternate interior Consecutive interior
∠
1 and
∠
5
∠
2 and
∠
8
∠
3and
∠
5
∠
4 and
∠
5
∠
4 and
∠
8
∠
1 and
∠
7
∠
4 and
∠
6
∠
3 and
∠
6
∠
2 and
∠
6
∠
3 and
∠
7
Same-Side Interior Angles:
two angles that lie between the two lines on the same side of the transversal.
∠
4
and
∠
5
are same-side interior angles.
(aka consecutive interior angles)
Assignment:
pp. 144-146; 12-22E, 25-28, 48.48.
3 2 Properties of Parallel Lines
Objectives:
I will be able to prove theorems about parallel lines.
I will be able to use properties of parallel lines to find angle measures.
Do Now 1. CE and AC are ______perpendicular _______. 2. EH and AB are _______skew _____________. 3. Plane DBG and plane AHE are _parallel _____. Use the figure below. Fill in the blank with parallel, skew, or perpendicular.
Postulate 3-1.
Same-Side Interior Angles Postulate
. If a transversal intersect two parallel lines, then same-side interior angles are supplementary.
If
l
∥
m
, then m
∠
4 + m
∠
5 = 180 m
∠
3 + m
∠
6 = 180
Theorem 3-1.
Alternate Interior Angles Theorem.
If a transversal intersects two parallel lines, then alternate interior angles are congruent.
If
l
∥
m
, then
∠
4
≅
∠
6
∠
3
≅
∠
5
Theorem 3-2.
Corresponding Angles Theorem.
If a transversal intersects two parallel lines, then corresponding angles are congruent. If
l
∥
m
, then
∠
1
≅
∠
5
∠
2
≅
∠
6
∠
3
≅
∠
7
∠
4
≅
∠
8
Theorem 3-3.
Alternate Exterior Angles Theorem.
If a transversal intersects two parallel lines, then alternate exterior angles are congruent.
If
l
∥
m
, then
∠
1
≅
∠
7
∠
2
≅
∠
8
Given:
a
∥
b
Statements Reasons
1. a
∥
b 1. Given
2.
∠ 1 ≅ ∠ 5
2. Corresponding angles are congruent.
3.
∠
7
≅ ∠ 5
3. Vertical angles are congruent.
4.
∠ 1 ≅ ∠
7
4. Transitive Property of Congruence.
Prove:
∠
1
≅
∠
7
Given:
a
∥
b
Statements Reasons
1. a
∥
b 1. Given
2.
∠ 1 ≅ ∠
2
2. Alternate exterior angles are congruent
3.
∠ 2 & ∠ 3 are sup.
3. Linear Pairs are supplementary.
4.
∠ 1 & ∠ 3
are sup.
4. Congruent Supplements Theorem
Prove:
∠
1
and
∠
3 are supplementary
Application.
Use properties of parallel lines to find the value of x. a. Alternate Exterior Angles Justify your answers by referring to specific Theorems. x - 8 = 55 x = 63 Linear Pairs b. Corresponding Angles (14x + 7) + 103 = 180 14x + 110 = 180 14x = 70 x = 5 c. Linear Pairs Corresponding Angles (4x
–
3) + 135 = 180 4x + 132 = 180 4x = 48 x = 12 d. Alternate Exterior Angles 9x + 7 = 115 9x = 108 x = 12
Assignment:
pp. 8-20E, 26.

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