GEOMETRY Chapter 1

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

NAMING FIGURES l FIGURE DESCRIPTION NAME IT A POINT A B D C 3 POINTS B, C, D A line containing 3 known points E F G l FE FG EF GE OR ..... OR l H J A segment with 2 end points HJ JH OR

NAMING FIGURES FIGURE DESCRIPTION NAME IT OR KL KM R, S, & T Ray with endpoint K OR KL KM M L K Collinear points R, S, & T R T S U Noncollinear points U, R, S, & T Opposite rays are two rays that share the same endpoint and they form a line. X Z Y and YX YZ

NAMING FIGURES NAME IT FIGURE DESCRIPTION Q NOP OR Q A plane containing 3 known points OR

DESCRIBING FIGURES Describe the figure: y O P Q R 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.

DISTANCE On a Number Line = 13 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. E F G -15 -2 7 The length of FG is | F – G |. FG = | F – G | = | -15 – - 2 | = | -15 + 2 | YOU TRY: Find GE and FE. = | -13| = 13

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

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

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

Example: Find the distance between 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, XW = 24 , WZ = 53 , Find XZ . X Z W 53 24 XW + WZ = XZ 24 + 53 = XZ 77 = XZ

Segment Addition Postulate If W is between X and Z, XW = 69 , XZ = 142, Find WZ . 142 X Z W 69 XW + WZ = XZ 69 + WZ = 142 – 69 – 69 WZ = 73

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

Segment Addition Postulate 4x + 6 3x + 26 = = 9x + 12 9x + 12 + 7x + 32 = 9x + 12 - 7x - 7x 32 = 2x + 12 - 12 - 12 20 = 2x 10 = x PG = 46 MG = 56 MP = 102 PG = 4x + 6 = 4(10) + 6 = 46 MG = 3x + 26 = 3(10) +26 = 56 MP = 9x + 12 = 9(10) + 12 = 102

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

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

Interior of an Angle

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

Congruent angles are angles Angle Measures are equal!! 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 . R P S T

Angle Addition Postulate 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 . 25° 60° K ?° m GHK + m KHI = m GHI 25 + x = 60 m KHI = 35 X = 60 – 25 = 35

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

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

If X is the midpoint of AB, Midpoints If X is the midpoint of AB, B X A 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. 12 47 - 23 -23 + 47 24 2 = = 12 2

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 60 - 14 23 x 2 - -14 = 46 - -14 = 46 +14 = 60

Midpoint Formula Midpoints on a Grid The Midpoint Formula: The midpoint of a segment with endpoints (x1 , y1) and (x2 , y2) has coordinates

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