Ch. 1 Essentials of Geometry

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Ch. 1 Essentials of Geometry

• Review

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Coplanar Objects
Remember Any 3 non-collinear points
determine a plane!

• Coplanar objects (points, lines, etc.) are
  objects that lie on the same plane. The plane
  does not have to be visible.

Are the following points coplanar?
A, B, C ?
Yes
A, B, C, F ?
No
H, G, F, E ?
Yes
E, H, C, B ?
Yes
A, G, F ?
Yes
C, B, F, H ?
No
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Front Side True or False
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Example 3

• Point S is between point R and point T. Use the
  given information to write an equation in terms
  of x. Solve the equation. Then find both RS and
  ST.
• RS 3x 16
• ST 4x 8
• RT 60

I---------------60 -------------------I
3x-16
I-----------4x-8-----------I
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EXAMPLE 2
Use algebra with segment lengths
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GUIDED PRACTICE
line l
Identify the segment bisector of .
Then find PQ.
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MIDPOINT FORMULA
The midpoint of two points P(x1, y1) and Q(x2,
y2) is M(X,Y) M(x1 x2, x2 y2)
2 2
Think of it as taking the average of the xs and
the average of the ys to make a new point.
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EXAMPLE 3
Use the Midpoint Formula
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EXAMPLE 3
Use the Midpoint Formula
SOLUTION
a. FIND MIDPOINT Use the Midpoint Formula.
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EXAMPLE 3
Use the Midpoint Formula
4 y 2
1 x 4
y 2
x 3
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Distance Formula

• The distance between two points A and B
• is

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EXAMPLE 4
Standardized Test Practice
SOLUTION
Use the Distance Formula. You may find it helpful
to draw a diagram.
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Naming Angles

• Name the three angles in diagram.

• Name this one angle in 3 different ways.

?WXY, ?WXZ, and ?YXZ
The vertex of the angle
What always goes in the middle?
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EXAMPLE 2
Find angle measures
SOLUTION
Angle Addition Postulate
Substitute angle measures.
145 6x 7
Combine like terms.
Subtract 7 from each side.
138 6x
Divide each side by 6.
23 x
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EXAMPLE 2
Find angle measures
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GUIDED PRACTICE
Find the indicated angle measures.
SOLUTION
Straight angle
Substitute angle measures.
Combine like terms.
Subtract 2 from each side.
Divide each side by 14.
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GUIDED PRACTICE
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EXAMPLE 3
Double an angle measure
SOLUTION
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Example 4
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EXAMPLE 2
Find measures of a complement and a supplement
a. Given that 1 is a complement of 2
and m 1 68, find m 2.
SOLUTION
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EXAMPLE 3
Find angle measures
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EXAMPLE 3
Find angle measures
SOLUTION
Write equation.
(4x 8) (x 2) 180
Substitute.
5x 10 180
Combine like terms.
5x 170
Subtract 10 from each side.
x 34
Divide each side by 5.
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EXAMPLE 3
Find angle measures
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Angles Formed by the Intersection of 2 Lines
? Click Me!
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EXAMPLE 4
Identify angle pairs
SOLUTION
To find linear pairs, look for adjacent angles
whose noncommon sides are opposite rays.
To find vertical angles, look or angles formed
by intersecting lines.
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Example 5

• Two angles form a linear pair. The measure of
  one angle is 5 times the measure of the other.
  Find the measure of each angle.

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Example 6

• Given that m?5 60? and m?3 62?, use your
  knowledge of linear pairs and vertical angles to
  find the missing angles.

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EXAMPLE 1
Identify polygons
Tell whether the figure is a polygon and whether
it is convex or concave.
SOLUTION
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of sides Type of Polygon
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
dodecagon
n-gon
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4
5
6
7
8
9
10
12
n
What is a polygon with 199 sides called?
199-gon
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EXAMPLE 2
Classify polygons
Classify the polygon by the number of sides. Tell
whether the polygon is equilateral, equiangular,
or regular. Explain your reasoning.
SOLUTION
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EXAMPLE 3
Find side lengths
SOLUTION
First, write and solve an equation to find the
value of x. Use the fact that the sides of a
regular hexagon are congruent.
Write equation.
Subtract 3x from each side.
Add 2 to each side.
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EXAMPLE 3
Find side lengths
Then find a side length. Evaluate one of the
expressions when x 8.
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Perimeter/Area

• Rectangle
• Square
• Triangle
• Circle

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Area

• The area of the triangle is 14 square inches and
  its height is 7 inches. Find the base of the
  triangle.

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Perimeter

• The perimeter of a rectangle 84.6 centimeters.
  The length of the rectangle is twice as long as
  its width. Find the length and width of the
  rectangle.