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

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

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  1. Math 010: Chapter 9GeometryLines, figures, & triangles November 25, 2013

  2. 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

  3. 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

  4. 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˚

  5. 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˚

  6. 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˚

  7. 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

  8. 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!

  9. Corresponding angles are congruent.

  10. 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˚

  11. Triangle equation • All angles in a triangle add up to 180˚ • Find C. • 38˚ + 85˚ + C = 180˚ • 123˚ + C = 180˚ • C = 57˚

  12. 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

  13. Types of triangles • Know what an isosceles, equilateral, scalene, and right triangle are. A right triangle has one right (90˚) angle.

  14. 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

  15. 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

  16. 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”

  17. Area of a rectangle

  18. Area of a triangle

  19. 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.

  20. 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

  21. Congruent triangles • Same size and shape – the exact same triangle • Rules to remember: ASA, SAS, SSS • Be able to identify which rule applies SAS

  22. 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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