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