GEOMETRY Chapter 1

CONTENTS Naming Figures Naming Figures Describing Figures Describing Figures Distance on a number line Distance on a number line Distance on a grid Distance on a grid Segment Addition Postulate Segment Addition Postulate Angles and Their Measures Angles and Their Measures Measuring Angles Measuring Angles Angle Addition Postulate Angle Addition Postulate Classify Angles Classify Angles Segment Bisectors and Midpoints Segment Bisectors and Midpoints Angle Bisectors Angle Bisectors

NAMING FIGURES FIGURE DESCRIPTION NAME IT APOINTA B D C 3 POINTS B, C, D A line containing 3 known points FE FG EF GE OR..... H J A segment with 2 end points HJJHOR E F GG

NAMING FIGURES FIGURE DESCRIPTION NAME IT Ray with endpoint K A plane containing 3 known points NOP Collinear points K M L ORKLKM N O P Q OR Q R, S, & T R T S U Noncollinear points U, R, S, & T

N O P Q R y Describe the figure: Plane Q contains Line NP, Line PR, and Points N, P, R, and O. Line NP and Line PR intersect at Point P. Line y intersects plane Q at point O. DESCRIBING FIGURES

DISTANCE On a Number Line E F G Finding the length of a segment is the same as finding the distance between its endpoints. When we measure a segment and attach a number to it we drop the bar in the symbol: Since the length of AB is 12, we write AB = 12. The length of FG is | F – G |. FG = | F – G | YOU TRY: Find GE and FE. = | -15 – - 2 | = | | = | -13| = 13

DISTANCE On a Number Line E F G The length of GE is | G – E |. GE = | G – E | = | - 2 – 7 | = | - 9| = 9 Find GE and FE. The length of FE is | F – E |. FE = | F – E | = | - 15 – 7 | = | - 22| = 22

DISTANCE On a Number Line The length of PQ is | P – Q |. PQ = | P – Q | = | 16 – - 4 | = | 20| = 20 Find the length of the segment that has endpoints with coordinates P(16) and Q(- 4).

DISTANCE On a Grid Subtract x-values To find the distance between two points on a Grid, use the Distance Formula: Subtract y-values Square the result Add the results Take the SQUARE ROOT and simplify

DISTANCE On a Grid Example: Find the distance between A( - 10, 4) and B( - 6, 1) AB = 5

DISTANCE On a Grid Find the distance between C( 7, - 3) and D( - 5, 2) CD = 13

Segment Addition Postulate If B is between A and C, A C B Then AB + BC = AC

Segment Addition Postulate If W is between X and Z, X Z W XW + WZ = XZ = XZ 77 XW = 24, WZ = 53, Find XZ.

Segment Addition Postulate If W is between X and Z, X Z W XW + WZ = XZ WZ 142 = WZ = XW = 69, Find WZ. XZ = 142, – 69

Segment Addition Postulate = MP Find all three segment measures. PG + MG P M G 4x + 6 9x x x + 6 3x x = If G is between P and M, PG = 4x + 6, MP = 9x + 12, and MG = 3x + 26,

Segment Addition Postulate 4x + 6 3x x = 7x = 9 x = 2 x = x 20 2 x = x = 1 0 PG = 4x + 6 = 4(10) + 6 = 46 MG = 3x + 26 = 3(10) +26 = 56 MP = 9x + 12 = 9(10) + 12 = PG = 46 MG = 56 MP = 102

Angles and Their Measures L J KJ and KL form  JKL sides  Angle symbol Naming angles (4 ways) 1)  JKL 2)  LKJ 3)  K (only if 1 angle) 4)  1 1 Vertex is K K K K

Naming Angles  MNP or  PNM  ONP or  PNO  MNO or  ONM

Interior of an Angle

Adjacent Angles  Common vertex  Common ray  No interior points in common

Measuring Angles Congruent angles are angles with the same measure. If m  ABC = 50  and m  JKL = 50  Then  ABC   JKL Angles are congruent Angle Measures are equal!! 

Angle Addition Postulate If P is in the interior of  RST, then m  RSP + m  PST = m  RST. TS R P

Suppose that the angle at the right measures 60° and that there is a point K in the interior of the angle such that m  GHK = 25 . Find m  KHI. m  GHK + m  KHI = m  GHI K 60° ?° 25° 25 + x = 60 X = 60 – 25 = 35 m  KHI = 35

Classify Angles Right Angle 90  Obtuse Angle 90  < x < 180  Straight Angle 180  Acute Angle 0  < x < 90 

Segment Bisector Bisect means to cut into 2 congruent pieces. The midpoint of a segment is the point that bisects the segment. A segment bisector is a segment, ray, line or plane that intersects the segment at its midpoint.

Construct the Midpoint of a Segment

Midpoints Midpoints If X is the midpoint of AB, A X B Then, AX = XB.

Midpoints on number lines To find the midpoint of a segment on a number line, just average the coordinates of the endpoints = =

Endpoints on number lines To find the endpoint of a segment on a number line with one endpoint and the midpoint: Midpoint x 2, then subtract the known endpoint = 23 x = = 60

Midpoint Formula Midpoints on a Grid The Midpoint Formula: The midpoint of a segment with endpoints (x 1, y 1 ) and (x 2, y 2 ) has coordinates

( ) Midpoint on a Grid A is (-3, 4) B is ( 2, 1) Midpoint is -3 +2, (-.5, 2.5)

Endpoint on a Grid A segment has endpoint J(-7, 8) and midpoint P(2, -1). Find the other endpoint. Double the midpoint P(2, -1). Then subtract the endpoint you know J(-7, 8). P(2, -1) x 2 gives (4, -2). (4, -2) - (-7, 8) (4 - -7, -2 – 8) (11, -10) The other endpoint is (11, -10).

Angle Bisectors An angle bisector is a ray that divides an angle into two adjacent congruent angles. Angle bisector

Construct an Angle Bisector