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 AD = length of the whole line segment

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 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 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 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˚

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 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 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 All angles in a triangle add up to 180˚ Find C.

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

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

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

Area of a circle 𝐴=𝜋 𝑟 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 The hypotenuse of a right triangle is the side opposite the right angle. Pythagorean Theorem: 𝑎 2 + 𝑏 2 = 𝑐 2 where c is the hypotenuse. Use this theorem with the “3-4-5” triangle 3 2 + 4 2 =9+16=25= 5 2 On exam, show this process to find the value of the hypotenuse.

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 SAS 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).