math 010: chapter 9 geometry lines, figures, & triangles november 25, 2013
TRANSCRIPT
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MATH 010: CHAPTER 9GEOMETRYLINES, FIGURES, & TRIANGLES
November 25, 2013
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9.1 Intro to Geometry (Lines & Angles)
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
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Know how to construct & solve this equation
If AD = 12 cm, AB = 5 cm, and CD = 4 cm, find the length of BC.
5cm x 4cm 5 + x + 4 = 12 x + 9 = 12 x = 3 Final Answer: BC = 3 cm
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Solve a supplementary angles equation
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˚
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Complementary angles equation
Complementary angles add up to 90˚ Solve for x. (x+3)˚ + (2x – 3)˚ = 90˚ x˚ +3˚ + 2x˚ – 3˚ = 90˚ 3x˚ = 90˚ x = 30˚
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Angles: Types of angles
1. Acute angles are smaller than 90 degrees Examples: 10˚, 45˚, 80˚
2. Right angles are 90 degrees Perpendicular 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˚
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Vertical angles are congruent 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
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Parallel lines and transversals 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!
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Corresponding angles are congruent.
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Know how to fill in all angle measures
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˚
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Triangle equation
All angles in a triangle add up to 180˚ Find C. 38˚ + 85˚ + C = 180˚ 123˚ + C = 180˚ C = 57˚
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9.2 Plane Geometric Figures
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.
pentagon
hexagon
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Types of triangles
Know what an isosceles, equilateral, scalene, and right triangle are.
A right triangle has one right (90˚) angle.
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Perimeter
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
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
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Area of a circle
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”
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Area of a rectangle
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Area of a triangle
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9.3 Triangles
The hypotenuse of a right triangle is the side opposite the right angle.
Pythagorean Theorem: where c is the hypotenuse.
Use this theorem with the “3-4-5” triangle
On exam, show this process to find the value of the hypotenuse.
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Similar triangles
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
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Congruent triangles
Same size and shape – the exact same triangle
Rules to remember: ASA, SAS, SSS Be able to identify which rule applies
SAS
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Quiz
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
If <1 = 60˚, find the measures of all other angles (2 through 8).