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Geometry CP
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Unit 3: Parallel and Perpendicular Lines Notes
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Back in chapter 2 we looked at what polygons
were.
The “official” definition of polygons is
threefold.
1. plane figures formed from three or more
segments.
2. Each segment intersects at exactly two
other segments.
3. No two segments with a common endpoint
are collinear.
3.1—Symmetry in
Polygons
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Segments of a polygon are called sides.
Common endpoints of a polygon are called
vertices.
Types of Polygons
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A 12-sided polygon is called a dodecagon.
All other polygons are called n-gons.
(i.e. a polygon with 15 sides is called a 15-gon.)
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Equiangular polygon—polygon where all
angles are congruent.
Equilateral polygon—polygon where all sides
are congruent.
Regular polygon—polygon where all sides and
all angles are congruent.
More Types of Polygons
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The center of a regular polygon is the point
that is equidistant from all vertices of the
polygon.
A central angle of a regular polygon is an angle
whose vertex is the center of the polygon and
whose sides pass through two consecutive
vertices.
Parts of a Polygon
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Take a piece of paper and fold it in half.
When you do this, you are creating reflectional
symmetry.
The crease that you make in the paper is called
the axis of symmetry.
Reflectional Symmetry
Types of
Triangles
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In your groups, predict how many axes of
symmetry the following triangles would have:
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Equilateral
Isosceles
Scalene
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Team Huddle
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Rotational symmetry exists if and only if a
polygon has at least one rotation image that
coincides with the original image.
Formula =
360
n
Rotational Symmetry
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In your teams, huddle up, take 60 seconds and
discuss everything you know about lines.
3.3—Parallel Lines and
Transversals
When you look at this picture, what do you know about:
angles 1 and 2?
angles 1 and 3?
angles 1 and 4?
Now, what is the difference between the picture in the
previous slide and this slide?
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The line that is being pointed to is called a
transversal.
A transversal is a line, ray, or segment that
intersects two or more coplanar lines, rays, or
segments, each at a different point.
Transversals
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When a transversal cuts two parallel lines,
eight angles are created.
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Types of Angles
Angle 3 and Angle 5
are alternate interior
angles.
Angle 1 and Angle 7
are alternate exterior
angles.
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Angle 3 and Angle 6
are same-side interior
angles.
Angle 1 and Angle 8
are same-side exterior
angles.
Angle 1 and Angle 5
are corresponding
angles.
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Angle 3 and Angle 5 are
alternate interior angles.
Is there another example?
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Angle 1 and Angle 7 are
alternate exterior angles.
Is there another example?
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Angles 3 and 6 are sameside interior angles. Is
there another example?
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Angles 1 and 5 are
corresponding angles. Is
there another example?
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If two lines cut by a transversal are parallel,
then corresponding angles are congruent.
Corresponding Angles
Postulate
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If two lines cut by a transversal are parallel,
then alternate interior angles are congruent.
Alternate Interior Angles
Theorem
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If two lines cut by a transversal are parallel,
then alternate exterior angles are congruent.
Alternate Exterior Angles
Theorem
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If two lines cut by a transversal are parallel,
then same-side interior angles are
supplementary.
Same-Side Interior Angles
Theorem
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In a team huddle, take 60 seconds to discuss
what you know about parallel lines.
3.4—Proving Parallel Lines are Like,
You Know, Parallel
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Remember from chapter 2 what a converse is.
What would the converse of this statement be?
“If two lines cut by a transversal are parallel,
then corresponding angles are congruent”?
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If two lines are cut by a transversal in such a
way that corresponding angles are congruent,
then the two lines are parallel.
Theorem: Converse of the
Corresponding Angles Postulate
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What do you know
about x?
What do you know
about the two lines?
What else can you
prove?
Using the Converses
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What do you know
about x?
What do you know
about the two lines?
What else can you
prove?
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In your team huddle, take 60 seconds and
discuss everything you can remember about
polygons.
3.6—Angles in Polygons
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In terms of the angles inside a polygon, they
can be described one of two ways.
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1. Convex Polygon—no part of a line segment
connecting any two points on the polygon is
outside the polygon.
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2. Concave Polygon—part of a line segment
connecting any two points on the polygon is
outside the polygon.
Other Types of Polygons
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In order to figure out the sum of the interior
angles of any polygon, we use the formula
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180(n-2)
Interior Angles of a
Polygon
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Based on the formula in the previous slide, can
you fill in the following table?
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Number of
Sides
Number of
Triangular
Regions
Sum of
Measures of
Angles
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In a regular polygon, all of the interior angles
measure the same.
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We can figure out the measure of one interior
angle by using the formula:
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180(n-2)
n
Measure of One Interior Angle of a
Regular Polygon
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Based on the formula in the previous slide, can
you fill in the following table?
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Number of
Sides
Sum of
Measures of
Interior
Angles
Measure of
One Interior
Angle
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The sum of the measures of the exterior angles
of a polygon is 360 degrees.
What else do you know about 360 degrees?
To determine the measure of one exterior angle,
we need to take 360/n
Sum of the Exterior
Angles of a Polygon
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Using the information provided in the previous
slide, huddle up and determine the measure of
one exterior angle of the following:
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Rhombus
Octagon
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Team Huddle
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So to figure out the sum of exterior and interior
angles of any polygon we have to use the
formula:
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180(n-2) + 360
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Using the information provided in the previous
slide, huddle up and determine the sum of the
exterior and interior angles of the following:
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Triangle
Pentagon
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Team Huddle