Lesson 1-R Chapter Review

Lesson Outline Five-Minute Check Objectives Vocabulary Core Concepts Examples Summary and Homework

5-Minute Check on Lesson 1-6 𝑩𝑫 bisects ABC. Use the diagram and the given angles to answer. mABD = 57°. Find mDBC and mABC. mABC = 110°. Find mDBC and mABD. B is supplement of A and mA = 65°. Find mB C is complement of A and mA = 60°. Find mC D is a linear pair with A and mA = 55°. Find mD E is vertical with A and mA = 45°. Find mE mDBC = 57° and mABC = 114° mDBC = 55° and mABD = 55° mB + mA = 180° 180 – 65 = 115° = mB mB + mA = 90° 90 – 60 = 30° = mC mD + mA = 180° 180 – 55 = 125° = mD mE = mA 45 = mE

Objectives Review chapter one material and vocabulary

Vocabulary none new Must know the terms from chapter 1 !!! Complementary versus supplementary Linear pairs versus vertical angles Concave versus convex Regular versus irregular Schwarzwaldekirchekȕchen means what?

Visual Definitions k Points A, B, C, D Line k Collinear A, B, C y A, B, C, D Line k (-5,5) k (6,4) D C Collinear (0,1) A, B, C x A Line Segments (-6,-2) BA, BC, AC B Plane xy coordinate Coplanar A, B, C, D

Whole = Sum of its Parts Any distance (or angle) can be broken into pieces and the sum of those pieces is equal to the whole distance (or measure of an angle) 11 14 6 A B C D 31 The whole length, AD, is equal to the sum of its parts, AB + BC + CD AD = AB + BC + CD 31 = 11 + 14 + 6

Distance Review Concept Formula Examples Distance Nr line Coord Plane D = | a – b | D = | 2 – 8| = 6 D = (x2-x1)2 + (y2-y1)2 D = (7-1)2 + (4-2)2 = 40 a b 2 3 4 5 6 7 8 9 1 (1,2) (7,4) Y X ∆x ∆y D Pythagorean Theorem on the formula sheet !!

Mid-points Review Concept Formula Examples Mid point Nr line Coord Plane (a + b) 2 (2 + 8) 2 [x2+x1] , [y2+y1] 2 2 7 + 1 , 4 + 2 2 2 = 5 = (4, 3) a b 2 3 4 5 6 7 8 9 1 (1,2) (7,4) Y X M Given an endpoint and the midpoint is a “travel” problem Endpoint: (1, 2) Midpoint: (4, 3) By equations: Midpoint (4, 3) Midpoint (4, 3) ̶ Endpoint (1, 2) + Travel (3, 1) Travel (3, 1) Other End (7, 4) By graphing: 3 right and 1 up to get to the middle. Have to go same distance to get to the other end.

Interior of angle AVB or V Angles 360º A Circle Exterior of angle Ray VA Interior of angle AVB or V Vertex (point V) V Ray VB B Angles measured in degrees A degree is 1/360th around a circle Acute Right Obtuse A A A mA < 90º mA = 90º 90º < mA < 180º Names of angles: Angles have 3 letter names (letter on one side, letter of the vertex, letter on the other side) like AVB or if there is no confusion, like in most triangles, then an angle can be called by its vertex, V

Angle Vocabulary Linear pair – a pair of adjacent angles whose noncommon sides are opposite rays (always supplementary) Vertical angles – two non adjacent angles formed by two intersecting lines. Vertical angles are congruent (measures are equal)!! Complementary Angles – two angles whose measures sum to 90° Supplementary Angles – two angles whose measures sum to 180° Perpendicular – two lines or rays are perpendicular if the angle (s) formed measure 90°

Figures Vocabulary Number of Sides Polygon Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n N-gon Concave – figure has an interior angle greater than 180 Convex – no sides “extended” pass through the interior of the figure Regular – all sides equal and all angles equal in a figure Irregular – sides or angles are different measures in a figure Perimeter – all sides of a figure added up Area – the “square units” of a figure’s interior

Summary & Homework Summary: Homework: Vocabulary is very important SOL topics: Distance formula Midpoints (regular and travel problems) Special angle pairs Complementary and supplementary Homework: study for the quiz

Vocabulary Important Complementary versus Supplementary Perpendicular Collinear versus Coplanar Perimeter Adjacent angles Vertical angles versus Linear Pairs Congruent Segments and Angles Concave versus Convex Regular versus Irregular Bisector (angle or segment)

Symbols are like Vocabulary What do the following symbols mean?     Congruent Segments Congruent Angles Right Angle m = 90 Angle Bisector Ray Line Segment Line Congruent Angle Perpendicular Equal

Finding Angles Information Once around a point is 360 Linear Pairs are supplementary Sum of triangle’s angles is 180 Vertical angles are congruent 3 = 151 5 = 151 4 = 29 6 = 29 12 = 42 10 = 42 11 = 138 9 = 138 7 = 109 1 = 109 2 = 71 8 = 71 1 2 3 4 5 6 7 8 10 11 9 12 If 3 = 151 and 12 = 42, find the rest

Finding Distance Distance formula and Pythagorean theorem are very similar Either can be used; formula must be memorized, but theorem is on formula sheet Given: A (2, 4) and B (5, 8), find the length of AB y x a = 5 – 2 = 3 b = 8 – 4 = 4 d = (a)2 + (b)2 d = (3)2 + (4)2 = (9) + (16) =  25 = 5 Given: C (-1, -4) and D (-5, -4), find the length of CD d = (-1 – (-5))2 + (-4 – (-4))2 d = (4)2 + (0)2 d = (16) = 4

Finding Midpoints Midpoint formula must be memorized Two problem types Given two endpoints, find the midpoint Given endpoint and midpoint, find the other endpoint Given: A (2, 4) and B (5, 8), find the midpoint of AB Given: A (-3, -5) and the midpoint of AB is (1, -1), find B x: (2+5)/2 = 7/2 = 3.5 y: (4+8)/2 = 12/2 = 6 Midpoint (3.5, 6) MP (1, -1) - EP (-3, -5) Travel (+4, +4) + MP (1, -1) OEP (5, 3) Travel -3 to 1 is + 4 -5 to -1 is + 4

Miscellaneous Perimeter is once around the figure; add up the sides Equations are the language of Math Any geometry problem can be turned into an algebra problem D A A D C B B C CAB = 2x – 6 , CAD = x + 8, Find x and BAD AB = 2x, AD = 2x + 8, BC = 3x – 2 Find x and perimeter 2x – 6 = x + 8 so x = 14 BAD = CAB + CAD = 3x + 2 = 3(14) + 2 = 44 P = 2x + 2x + 8 + 2x + 3x – 2 = 9x + 6 = 96 2x + 8 = 3x – 2 so x = 10