MATH 010: CHAPTER 9
GEOMETRY
LINES, FIGURES, & TRIANGLES
November 25, 2013
9.1 Intro to Geometry (Lines & Angles)
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Lines have infinite length, they go on forever
Line segments have a finite length
The length of a segment is denoted by the two
endpoints. AB = distance between A and B
AD = length of the whole line segment
Know how to construct & solve this equation
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If AD = 12 cm, AB = 5 cm, and CD = 4 cm, find the
length of BC.
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5cm
5 + x + 4 = 12
x + 9 = 12
x=3
Final Answer: BC = 3 cm
x
4cm
Solve a supplementary angles equation
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180˚ is a straight line
Supplementary angles add up to 180˚
Think straight = supplementary
What is the value of b?
45˚ +39 ˚ + b + 24˚ = 180˚
b + 108 = 180
b = 72˚
Complementary angles equation
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Complementary angles add up to 90˚
Solve for x.
(x+3)˚ + (2x – 3)˚ = 90˚
x˚ +3˚ + 2x˚ – 3˚ = 90˚
3x˚ = 90˚
x = 30˚
Angles: Types of angles
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1. Acute angles are smaller than 90 degrees
 Examples:
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10˚, 45˚, 80˚
2. Right angles are 90 degrees
 Perpendicular
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lines are lines that form a right angle
3. Obtuse angles are larger than 90 degrees and
smaller than 180 degrees
 Examples:
100˚, 160˚, 95˚
Vertical angles are congruent
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Congruent angles have equal measure.
Vertical angles are the angles formed across from
each other by two intersecting lines.
Also note that 134˚ and 46˚ are supplementary
Parallel lines and transversals
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Parallel lines are lines that will never intersect no
matter how long you draw them.
A transversal is a line that intersects two other lines
at different points
Alternate interior angles are shown here:
 AIA’s
are congruent!
Corresponding angles are congruent.
Know how to fill in all angle measures
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Given: <1 measures 110˚
Note that <1 and <2 are supplementary
So <2 measures 70˚
All angles in this picture measure either 110˚ or 70˚
Triangle equation
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All angles in a triangle add up to 180˚
Find C.
38˚ + 85˚ + C = 180˚
123˚ + C = 180˚
C = 57˚
9.2 Plane Geometric Figures
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Polygons are shapes made up
of 3 or more line segments:
triangles, rectangles, octagons,
etc.
Circles, ovals are not polygons.
A regular polygon is a
polygon where all sides are
equal, and all angles are
equal.
Know this: a pentagon has 5
sides. A hexagon has 6 sides.
hexagon
pentagon
Types of triangles
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Know what an isosceles, equilateral, scalene, and
right triangle are.
A right triangle has one right (90˚) angle.
Perimeter
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The perimeter is the distance around the outside of
a figure.
To find the perimeter of a polygon, add up all the
side lengths.
Perimeter of this rectangle
= 2 cm + 6 cm + 2 cm + 6 cm = 16 cm
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Circumference
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Circumference is the distance around a
circle.
C = 2πr or πd
Find the circumference of a circle with
diameter 10.
Circumference = 10 π
Find the circumference of a circle with
radius 2.
Circumference = 2π2 = 4π
Area of a circle
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𝐴 = 𝜋𝑟 2
First need to square r (order of operations)
Find the area of a circle with radius 5.
5 squared is 25
A = 25π
Remember the two circle formulas
Area is the one containing “squared”
Area of a rectangle
Area of a triangle
9.3 Triangles
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The hypotenuse of a right triangle is the
side opposite the right angle.
Pythagorean Theorem: 𝑎2 + 𝑏2 = 𝑐 2
where c is the hypotenuse.
Use this theorem with the “3-4-5” triangle
32 + 42 = 9 + 16 = 25 = 52
On exam, show this process to find the
value of the hypotenuse.
Similar triangles
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Similar means same shape
Does not mean same size
Angle measures same
Side lengths proportional
Know how to find missing side
Multiplication
We know 14 = 7 · 2;
12 = 6 · 2
So, 10 · 2 = 20
Congruent triangles
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Same size and shape – the exact same triangle
Rules to remember: ASA, SAS, SSS
Be able to identify which rule applies
SAS
Quiz
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Overall, rate how confident you
feel (1-5, 5 best) about the
following:
Geometry vocab
 Lines and angles equations
 Area formulas
 Similar triangles (proportion)
 Congruent triangles rules
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If <1 = 60˚, find the measures of
all other angles (2 through 8).