Download Geometry Chapter 3 Parallel Lines and Planes

Geometry Chapter 3
Parallel Lines and
Perpendicular Lines
Pages 124 195
3-1 PAIRS & LINES OF ANGELS
What you will learn:
 Identify lines and planes
 Identify parallel and
perpendicular lines
 Identify pairs of angles
formed by transversals
3-1 PROPERTIES OF PARALLEL LINES
Essential Question:
What does it mean when two
lines are parallel,
intersecting, coincident, or
skew?
PREVIOUS VOCABULARY
 Perpendicular lines
CORE VOCABULARY
 Parallel Lines
 Skew Lines
 Parallel Planes
 Transversal
 Corresponding
Angles
 Alternate interior
Angles
Angles
 Same-Side
(consecutive)
interior angles
PARALLEL LINES
 Two lines
that do not
intersect
 Go in same
direction
 Coplanar
SKEW LINES
 Two lines
that do not
intersect
 Are not
coplanar
PARALLEL PLANES
 Two planes
that do not
intersect
TRANSVERSAL
 A line that
intersects
two or more
coplanar
parallel lines
CORRESPONDING
ANGLES
 Congruent
 Same
position
 Different
location
ALTERNATE INTERIOR
ANGLES
 Congruent
 Inside
 Opposites
sides
ALTERNATE EXTERIOR
ANGLES
 Congruent
 Outside
 Opposites
sides
SAME-SIDE (consecutive)
INTERIOR ANGLES
 Supplementary
 Inside
 Same side
PARALLEL LINES
Two
coplanar
lines that
do not
intersect
STRAIGHT ANGLE
Exactly 180 degrees
VERTICAL ANGLES
2 angles directly across
from each other
congruent
SUPPLEMENTARY
ANGLES
Two angles whose
measures add up to 180
degrees
3 – 2 PARALLEL LINES &
TRANSVERSALS
What you will learn:
 Use properties of parallel lines
 Prove theorems about parallel
lines
 Solve real-life problems
3-2 PARALLEL LINES & TRANSVERSALS
Essential Question:
 When two parallel lines are
cut by a transversal, which
of the resulting pairs of
angles are congruent?
CORE VOCABULARY
 Transversal
 Corresponding
Angles
 Alternate interior
Angles
 Alternate
Exterior Angles
 Same-Side
(consecutive)
interior angles
TRANSVERSAL
 A line that
intersects
two or more
coplanar
parallel lines
CORRESPONDING
ANGLES
 Congruent
 Same
position
 Different
location
ALTERNATE INTERIOR
ANGLES
 Congruent
 Inside
 Opposites
sides
ALTERNATE EXTERIOR
ANGLES
 Congruent
 Outside
 Opposites
sides
SAME-SIDE (consecutive)
INTERIOR ANGLES
 Supplementary
 Inside
 Same side
3 – 3 Proofs and Parallel Lines
What you will learn:
 Use the Corresponding Angles
Converse
 Construct Parallel Lines
 Prove theorems about parallel lines
 Use Transitive Property of Parallel
Lines
3 – 3 Proofs and Parallel Lines
Essential Question:
 Name the two types of pairs
of angles that are
supplementary
WAYS TO PROVE TWO LINES
PARALLEL
 Show that a pair of corresponding
angles are congruent
 Show that a pair of alternate
interior or exterior angles are
congruent
 Show that a pair of same-side
interior angles are supplementary
WAYS TO PROVE TWO LINES
PARALLEL
 Show that both lines are
perpendicular to a third line
 Show that both lines are parallel
to a third line
Core Concept:
Five Types of Angle Pairs
 Corresponding
≅
 Alternate Interior ≅
 Alternate Exterior ≅
 Same-Side Interior 180
 Vertical ≅
 Linear Pair 180
PERPENDICULAR LINES
Two lines that intersect
to form right angles
If a line is
perpendicular to one of
two parallel lines, it is
also perpendicular to
3 - 4 PROOFS WITH
PERPENDICULAR LINES
What you will learn:
 Find the distance from a point to a
line
 Construct Perpendicular lines
 Prove theorems about perpendicular
lines
 Solve real life problems involving
perpendicular lines
3 – 4 Proofs and Parallel Lines
Essential Question:
 What conjectures can you
make about perpendicular
lines?
VOCABULARY
 Distance from a point to a line
 Perpendicular bisector
Distance from a point to a line
 The length of the perpendicular
segment from the point to the
line
Perpendicular Bisector
 A perpendicular bisector of a line
segment is a line segment that is
perpendicular to the segment at its
midpoint
PARALLEL LINES
 Two lines that do not
intersect
 Go in same direction
 If two lines are parallel
to the same line, they
are parallel to each
other
 If two lines are
perpendicular to the
TRIANGLE
 Three sides
 Interior angle sum is
180˚
 Symbol: ∆
 Sides are called
segments
 Each point is a vertex
EQUIANGULAR
All angles are 60˚
ACUTE TRIANGLE
Three
angles less
than 90
degrees
RIGHT TRIANGLE
One right
angle
OBTUSE TRIANGLE
One obtuse
angle
EQUILATERAL TRIANGLE
All sides
congruent
ISOSCELES TRIANGLE
At least two
congruent sides
SCALENE TRIANGLE
No congruent
sides
EXTERIOR ANGLE
 Outside the triangle
 Equals the remote interior
angles
 Supplementary to its adjacent
angle
REMOTE INTERIOR
ANGLES
 on the opposite
side of the
exterior angles
 equal the
measure of the
exterior angle
3 - 5 POLYGON ANGLESUM THEOREM
STANDARD:
 classify polygons
 find measures of
interior and exterior
angles of polygons
VOCABULARY
1. Polygon
2. Concave Polygon
3. Convex Polygon
4. Diagonal
5. Polygon Angle Sum
6. Polygon Exterior Angle Sum
7. Equilateral Polygon
8. Equiangular Polygon
POLYGON
 Closed plane figure
 At least 3 sides and angles
 Classified by the number of
sides
CONVEX POLYGON
Doesn’t cave in
CONCAVE POLYGON
caves in
Diagonal
Connects vertices
POLYGON ANGLE SUM
(n-2)180
POLYGON EXTERIOR
ANGLE SUM
The exterior angles of
a polygon = 360
EQUILATERAL
POLYGON
All sides are
congruent
EQUIANGULAR
POLYGON*
All angles are
congruent
REGULAR POLYGON
Equiangular
Equilateral
3 - 6 LINES IN THE
COORDINATE PLANE
STANDARD:
 graph lines given their
equations
 to write equations of
lines
VOCABULARY
1. Slope
2. y-intercept
3. x-intercept
4. Graphing Using Intercepts
5. Standard Form
6. Slope Intercept Form
7. Point Slope Form
SLOPE
y-intercept
Where the graph
intersects the y-axis
x-intercept
Where the graph
intersects the x-axis
Graphing Using intercepts
Substitute “0” for x
and y to find the
intercepts
STANDARD FORM
Ax + By = C
SLOPE INTERCEPT FORM
y = mx + b
b = y-intercept
m = slope
POINT SLOPE FORM
y - y1 = m(x - x1)
3 - 7 SLOPES OF
PARALLEL AND
PERPENDICULAR LINES
STANDARD:
 relate slope and parallel
lines
 relate slope and
perpendicular lines
PARALLEL LINES
 Have equal
slopes
 Two lines
that do not
intersect
 Go in same
direction
PERPENDICULAR LINES
The product of slopes
is -1
Two lines that intersect
to form right angles
SLOPE INTERCEPT FORM
y = mx + b
b = y-intercept
m = slope
INTERSECT
To cut
Divide by passing
through
CONGRUENT
equal
The same