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Math 010: Chapter 9 Geometry Lines, figures, & triangles

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Math 010: Chapter 9 Geometry Lines, figures, & triangles. November 25, 2013. 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

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9 1 intro to geometry lines angles
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
know how to construct solve this equation
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
solve a supplementary angles equation
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˚
complementary angles equation
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˚
angles types of angles
Angles: Types of angles
  • 1. Acute angles are smaller than 90 degrees
    • Examples: 10˚, 45˚, 80˚
  • 2. Rightangles 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˚
vertical angles are congruent
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
parallel lines and transversals
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!
know how to fill in all angle measures
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˚
triangle equation
Triangle equation
  • 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
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.

hexagon

pentagon

types of triangles
Types of triangles
  • Know what an isosceles, equilateral, scalene, and right triangle are.

A right triangle has one right (90˚) angle.

perimeter
Perimeter
  • The perimeteris 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

circumference
Circumference
  • Circumference is the distance around a circle.
  • C = 2πr or πd
  • Find the circumference of a circle with diameter 10.
  • Circumference = 10π
  • Find the circumference of a circle with radius 2.
  • Circumference = 2π2 = 4π
area of a circle
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”
9 3 triangles
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.
similar triangles
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
congruent triangles
Congruent triangles
  • Same size and shape – the exact same triangle
  • Rules to remember: ASA, SAS, SSS
  • Be able to identify which rule applies

SAS

slide22
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).
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