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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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Math 010 chapter 9 geometry lines figures triangles

Math 010: Chapter 9GeometryLines, figures, & triangles

November 25, 2013


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


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).