Points, Lines, and Planes 1.1

Transcription

1 Points, Lines, and Planes 1.1 Point a location ex. write as: Line made up of points and has no thickness or width. ex. c write as:, line c ollinear points on the same line. Noncollinear points not on the same line. Plane flat surface made up of points; has no depth and extends infinitely in all directions. ex. X Write as: plane, plane X xample 1 Use the figure to answer. oplanar points that lie on the same plane. Noncoplanar points that do not lie on the same plane. 1a. Name a line containing point K. b. Name a plane containing point L. c. re points J, K, and M collinear? J K L a M d. re points K, L, and M coplanar? xample 2 Name the geometric shape. 1. The paper you are writing on. 2. The tip of your pencil/pen. 3. The track from the ceiling tile.

2 xample 3 raw and label a figure for each. 5. Plane R contains lines and which intersect at point P. dd point which is noncollinear to lines and. 6. QR on a coordinate plane contains Q (-2, 4) and R (4, -4). dd point T so that T is collinear with these points. xample 4 Use the figure to answer. 7. How many planes appear in the figure? 8. Name 3 collinear points. 9. re points,,, and coplanar? S 10. o and intersect? If so, where? Worksheet 1.1

3 istance and Midpoints 1.3 istance between two points a. Number Line b. oordinate Plane d = ( x - x ) + ( y - y ) xample 1 Use the number line to find GH. G H xample 2 Find the distance between (-4, 1) and F(3, -1). Midpoint (Segment) the point on the segment that divides the segment into two congruent segments. If X is the midpoint of, then X X. a. Number Line + 2 b. oordinate Plane x 1 + x , y y 2 2 xample 3 Find the coordinates of the midpoint if (-6, 4) and (8, 1). xample 4 Find the coordinates of W if X(3, -5) is the midpoint of WY and Y has coordinates of (12, 1).

4 xample 5 What is the measure of PR if Q is the midpoint of PR. P Q 14x x R xample 6 What is the measure of if is the midpoint of? 4x - 1 2x + 5 Segment isector any segment, line, or plane that intersects a segment at its midpoint. bisects Worksheet 1.3

5 ngle Measure 1.4 egree a unit of measurement used in measuring angles and arcs. n arc of a circle with a measure of 1 is 1 of the entire circle. 360 Ray part of a line that has one endpoint and extends infinitely in one direction.,, or ***Two opposite and collinear rays form a line*** ngle formed by two noncollinear rays that have a common endpoint. 2 parts: side rays vertex common endpoint Y X 3 X, YXZ, ZXY, or 3 Z Other distinct parts: Y X 3 Z Use the figure to answer. 1. Name all angles that have as a vertex. G Name the sides of Write another name for 6. F 3 4 ngle lassifications 1. Right an angle measuring exactly cute an angle measuring greater than 0 and less than < x < Obtuse an angle measuring greater than 90 and less than < x < 180

6 Use the figure to answer. U V T W S Y X pproximate each angle and classify. 4. TYV 5. WYT 6. TYU 60 F 60 F Solve each angle. 7. 3x + 5 5x - 25 ngle isector a ray that divides an angle into two congruent angles. S V T U Worksheet 1.4

7 ngle Relationships 1.5 djacent ngles two angles that lie in the same plane, have a common vertex and a common side, but no common interior points. Vertical ngles two nonadjacent angles formed by two intersecting lines. and Linear Pair a pair of adjacent angles with non common sides that are opposite rays. xample 1 Use the figure to answer. a. Name adjacent angles. b. Name vertical angles. F c. Name linear pairs. omplementary ngles two angles with measures that have a sum of F Supplementary ngles two angles with measures that have a sum of

8 xample 2 Find the measure of two complementary angles if the difference of the two angles is 16. xample 3 Find the measure of two supplementary angles if the measure of one angle is 6 less than 5 times the measure of the other angle. Perpendicular lines, segments, or rays that form right angles. xample 4 Find x if KO HM L K 3x + 6 I 9x J M N O H xample 5 etermine if each statement can be justified from the figure. a. VYT = 90 b. TYW and TYU are supplementary. c. VYW and TWS are adjacent angles. U Y X W V T S Worksheet 1.5

9 lgebraic Proofs 2.6 lgebra Properties Reflexive Property: Symmetric Property: Transitive Property: ddition/subtraction: Multiplication/ivision: Substitution Property: istributive Property: a = a If a = b, then b = a If a = b and b = c, then a = c If a = b, then a + c = b + c a c = b c If a = b, then a c = b c a b = c c If a = b, then a may be replaced by b a(b + c) = ab + ac Name the property illustrated by each statement. a. If 5x 20, then x 4. b. If 12, then 12. c. If, and, then.. d. If y 75, and m y, then m 75. e. If 4 x 2 5, then 4 x 3.. f. If 4(3x 4) 12 x 16 F. Solve 2(5 3c) 4(a + 7) = 92 Statements Reasons 1. 2(5 3a) 4(a + 7) = a 4a 28 = ddition Property ivision Property 6. a = Write a 2 column proof to show that if 7 d + 3 = 6, then d = 3. 4 Statements Reasons

10 Segment and ngles are real numbers; therefore properties of real numbers can apply. 3. Given: = 16, = Prove: = 16 Statements Reasons Worksheet 2.6

11 Parallel Lines coplanar lines that do not intersect. Parallel Lines and Transversals 3.1 Skew Lines lines that do not intersect and are not coplanar. xample 1 Use the figure to answer. a. Name a plane parallel to plane H. b. Name all segments parallel to G. F H G c. Name all segments that intersect. d. Name all segments that are skew to. Transversal a line that intersect two or more lines in a plane at different points. t Transversals and ngles xterior ngles angles on the outside of the lines. Interior ngles angles on the inside of the lines. onsecutive Interior ngles angles on the same side of the transversal and next to each other. lternate Interior ngles interior angles on opposite sides of the transversal and not adjacent. lternate xterior ngles - exterior angles on opposite sides of the transversal and not adjacent. orresponding ngles non-adjacent angles, 1 interior and 1 exterior, on the same side of the transversal. xample 2 Use the figure to answer. Identify each pair of angles and and and and and and 10 Worksheet 3.1

12 ngles and Parallel Lines 3.2 The following postulate and theorems have the same hypothesis, but a different conclusion. If two parallel lines are cut by a transversal, then: Postulate 3.1 each pair of corresponding angles is congruent. Theorem 3.1 each pair of alternate interior angles is congruent. Theorem 3.2 each pair of consecutive interior angles is supplementary. Theorem 3.3 each pair of alternate exterior angles is congruent. xample 1 In the figure x y and 11 = 51, find x y Theorem 3.4 In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. m n xample 2 What is m 1? t xample 3 If m 5 = 2x 10, m 6 = 4(y 25) and m 7 = x + 15, find x and y Worksheet 3.2