Geometry. Chapter 1 Points, Lines, Planes, and Angles

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1 Geometry Chapter 1 Points, Lines, Planes, and Angles ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** Algebraic Equations Review Keystone Vocabulary Writing Assignment (1-1) Points, Lines, and Planes Page 9-10 # 13-20, 21, 22, 25, 26, 27, (1-2) Linear Measure Day 1 Page 17 # 12-15, (1-2) Linear Measure Day 2 Page # 28-39, 58, 62 (1-3) Distance Day 1 Page 25 # (1-3) Midpoints Day 2 Page 26 # Take Home Test on Sections 1-1 through 1-3 (1-4) Angle Measure Day 1 Page # 4-6, (1-4) Angle Measure Day 2 Page 34 # (1-4) Angle Measure Day 3 Page 34 # 34 39, 52-60, 61, 63, 65 (1-5) Angle Relationships Day 1 Page 42 # (skip# 17) (1-5) Angle Relationships Day Practice WS (1-5) Angle Relationships Day 3 Page 42 # 17, 27-30, 31-35, 37 (1-6) Polygons Day 1 Page # 12-25, (1-6) Polygons Day 2 Page #26 28 Chapter 1 Review WS 1

2 Section 1 1: Points, Lines, and Planes Notes Date: A Point: is simply a. Drawn as a. Named by a letter. Words/Symbols: Example: A Line: is made up of and has no thickness or. Drawn with an at each end. Named by the representing two points on the line or a lowercase script letter. Points on the same are said to be. Words/Symbols: Example: A Plane: is a surface made up of. Drawn as a 4-sided figure. Named by a script letter or by the letters naming three points. Points that lie on the same plane are said to be. Words/Symbols: Example: 2

Example #1: Use the figure to name each of the following. a.) Name a line that contains point P. b.) Name the plane that contains lines n and m. c.) Name the intersection of lines n and m. d.

3 Example #1: Use the figure to name each of the following. a.) Name a line that contains point P. b.) Name the plane that contains lines n and m. c.) Name the intersection of lines n and m. d.) Name a point not on a line. e.) What is another name for line n. f.) Does line l intersect line n or line m? Explain. Example #2: Draw and label a figure for the following relationship. a.) Point T lies on WR. b.) AB intersects CD in plane Q at point P. Example #3: a.) How many planes appear in this figure? b.) Name three points that are collinear. c.) Are points A, B, C, and D coplanar? Explain. suur suur d.) At what point do DB and CA intersect? 3

4 CRITICAL THINKING 1.) Why do chairs sometimes wobble? Include the following in your answer: an explanation of how the chair legs relate to points in a plane, and how many legs would create a chair that does not wobble. 2.) Complete the figure below to show the following relationship: Lines a, b, and c are coplanar and lie in plane Q. Lines a and b intersect at point P. Line c intersects line b at point R, but does not intersect line a. 4

5 Date: Section 1 2: Linear Measure Notes Part 1 Measure Line Segments A line segment, or, is a measurable part of a line that consists of two points, called, and all of the points between them. A segment with endpoints A and B can be named as or. The length or of AB is written as. Example #1: Use a metric ruler to draw each segment. g.) Draw LM that is 42 millimeters long. b.) Draw QR that is 5 centimeters long. Example #2: Use a customary ruler to draw each segment. a.) Draw DE that is 3 inches long. b.) Draw FG that is inches long. 5

6 Calculate Measures Betweenness of Points: Point M is between points P and Q if and only if P,Q, and M are and. Example #4: a.) Find LM. b.) Find XZ. c.) Find DE. d.) Find x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x 3. e.) Find y and PQ if P is between Q and R, PQ = 2y, QR = 3y + 1, and PR = 21. Draw a picture! 6

7 Date: Section 1 2: Linear Measure Notes Part 2 Example: Find the value of x and LM if L is between N and M, NL = 6x 5, LM = 2x + 3, and NM = 30. Draw a picture! Measure Line Segments Key Concept (Congruent Segments): Two having the same measure are. Symbol: Ex: Example #1: Name all of the congruent segments found in the kite. 7

8 Example #2: Find the measurement of RS. Example #3: Use the figures to determine whether each pair of segments is congruent. a.) AB, CD b.) WZ, XY c.) HO, HT d.) MH, TH 8

9 CRITICAL THINKING 1.) Explain the difference between a line and a line segment and why one of these can be measured, while the other cannot. 2.) Refer to the figure to the right. a.) Name three collinear points. b.) Name two planes that contain points B and C. c.) Name another point in plane DFA. d.) How many planes are shown? 9

10 Date: Section 1 3: Distance Notes Part 1 Distance Between Two Points Key Concept (Distance Formulas): Number Line Coordinate Plane The distance d between two points with coordinates (x1, y1) and (x2, y2) is given by d = Example #1: Find the distance between E(-4, 1) and F(3, -1). 10

11 Example #2: Use the number line to find QR. Example #3: Use the number line to find CD. Example #4: Use the number line to find AB and CD. Example #5: Use the Distance Formula to find the distance between the following points. a.) A(10, -2) and B(13, -7) b.) X(-5, -7) and Y(-10, 7) c.) G(-4, 1) and H(3, -1) 11

12 Date: Section 1 3: Midpoint Notes Part 2 Midpoint of a Segment Key Concept (Midpoint): The midpoint M of PQ is the point P and Q such that. Number Line: The coordinate of the midpoint of a whose endpoints coordinates a and b is have Example #1: The coordinates on a number line of J and K are 12 and 16, respectively. Find the coordinate of the midpoint of JK. Hint: Draw a number line! Example #2: The coordinates on a number line of T and S are 5 and 8, respectively. Find the coordinate of the midpoint of TS. Hint: Draw a number line! Coordinate Plane: The coordinates of the of a segment whose endpoints have coordinates (x1, y1) (x2, y2) are and 12

13 Example #3: Find the coordinates of the midpoint of PQ for P(-1, 2) and Q(6, 1). Example #4: Find the coordinates of the midpoint of GH for G(8, -6) and H(-14, 12). Example #5: Find the coordinates of the midpoint of AB for A(4, 2) and B(8, -6). Example #6: What is the measure of PR if Q is the midpoint of PR? Segment Bisector: any segment, line, or plane that interests a segment at its 13

14 CRITICAL THINKING 1.) Which equation represents the following problem? Fifteen minus three times a number equals negative twenty-two. Find the number. a.) 15 3n = -22 b.) 3n 15 = -22 c.) 3(15 n) = -22 d.) 3(n 15) = ) Find the distance between points at (6, 11) and ( -2, -4). 14

15 Date: Section 1 4: Angle Measure Notes Part 1 Measure Angles Degree: a unit of measure used in measuring and. An arc of a circle with a measure of 1 is of the entire circle. Ray: is a part of a It has one and extends indefinitely in direction. Symbols: Opposite Rays: two rays and such that B is between A and C Key Concept (Angle): An angle is formed by two rays that have a common. The rays are called of the angle. The common endpoint is the. Symbols: 15

An angle divides a plane into three distinct parts. Points,, and lie on the angle. Points and lie in the interior of the angle. Points and lie in the exterior of the angle. Example #1: a.

16 An angle divides a plane into three distinct parts. Points,, and lie on the angle. Points and lie in the interior of the angle. Points and lie in the exterior of the angle. Example #1: a.) Name all angles that have B as a vertex. b.) Name the sides of 5. c.) Write another name for 6. Example #2: a.) Name all the angles that have W as a vertex. b.) Name the sides of 1. c.) Write another name for WYZ. d.) Name the vertex of 4. 16

17 Date: Section 1 4: Angle Measure Notes Part 2 Measure Angles Key Concept (Classify Angles): RIGHT ANGLE: ACUTE ANGLE: OBTUSE ANGLE: Model: Model: Model: Measure: Measure: Measure: Example #1: Measure each angle, then classify as right, acute, or obtuse. a.) b.) c.) d.) 17

18 e.) f.) Example #2: Measure each angle named and classify it as right, acute, or obtuse. a.) TYV b.) WYT c.) TYU d.) VYX e.) SYV 18

Arcs on the figure also indicate which angles are.

19 Date: Section 1 4: Angle Measure Notes Part 3 Congruent Angles Key Concept (Congruent Angles): Angles that have the same are congruent angles. Arcs on the figure also indicate which angles are. Example #1: State whether each pair of angles is congruent, and if so write a congruence statement. a.) b.) Example #2: Find the value of x and the measure of one angle. 19

) If m RQT = 6 x + 5 and m SQT = 7x 2, find m RQT. RQS. b.) Find m TQS if m RQS = 22a 11 and m RQT = 12a 8.

20 Angle Bisector: a that divides an angle into congruent angles. Ex: If uuur PQ is the angle bisector of, then. Example #3: In the figure, QP and QR are opposite rays, and QT bisects a.) If m RQT = 6 x + 5 and m SQT = 7x 2, find m RQT. RQS. b.) Find m TQS if m RQS = 22a 11 and m RQT = 12a 8. Example #4: In the figure, YU bisects a.) If m 1 = 5x + 10 and m 2 = 8x 23, find m 2. ZYW and YT bisects XYW. b.) If m WYZ =82 and m ZYU = 4 r + 25, find r. 20

21 CRITICAL THINKING 1.) Mr. Lopez wants to cover the walls of his unfinished basement with pieces of plasterboard that are 8 feet high, 4 feet wide, and ¼ inch thick. If the basement measures 24 feet wide, 16 feet long, and 8 feet tall, how many pieces of plasterboard will he need to cover all four walls? 2.) Each figure below shows noncollinear rays with a common endpoint. a.) Count the number of angles in each figure. b.) Describe the pattern between the number of rays and the number of angles. c.) Make a conjecture of the number of angles that are formed by 7 noncollinear rays and by 10 noncollinear rays. d.) Write a formula for the number of angles formed by n noncollinear rays with a common endpoint. 21

22 Section 1 5: Angle Relationships Notes Part 1 Date: Pairs of Angles Key Concept (Angle Pairs): Adjacent Angles: are two angles that lie in the same, have a common, and a common, but no common interior Examples: Vertical Angles : are two non-adjacent angles formed by two lines Examples: Non-example: Linear Pair : a pair of angles whose non-common sides are opposite. Example: Non-example: 22

23 Example #1 : Name an angle pair that satisfies each condition. a.) two angles that form a linear pair b.) two acute vertical angles c.) an angle supplementary to VZX d.) two acute adjacent angles Key Concept (Angle Relationships): Complementary Angles: two angles whose measures have a sum of Examples: Supplementary Angles: two angles whose measures have a sum of. Examples: Example #2: Find the measures of two supplementary angles if the measure of one angle is 6 less than 5 times the measure of the other angle. Example #3: Find the measures of two complementary angles if the difference in the measures of the two angles is 12. Example #4: The measure of an angle s supplement is 33 less than the measure of the angle. Find the measure of the angle and its supplement. 23

Perpendicular lines intersect to form angles. and can be perpendicular to lines or to other line segments and rays.

24 Section 1 5: Angle Relationships Notes Part 2 Date: Perpendicular Lines Lines that form right angles are. Key Concept (Perpendicular Lines): Perpendicular lines intersect to form right angles. Perpendicular lines intersect to form angles. and can be perpendicular to lines or to other line segments and rays. The right angle symbol in the figure indicates that the lines are. Symbol: is read is perpendicular to. suur suuur Example #1: Find x so that KO HM. Example #2: Find x and y so that BE and AD are perpendicular. 24

25 Assumptions: Example #3: Determine whether or not each of the following statements can be assumed or not. All points shown are coplanar. P is between L and Q. PN PL QPO and OPL are supplementary. PN PM L, P, and Q are collinear. QPO LPM PQ PO LP PQ LMP and MNP are adjacent angles. LPN and NPQ are a linear pair. OPN LPM PM, PN, PO, and LQ intersect at P. Example #4: Determine whether each statement can be assumed from the figure below. Explain. a.) m VYT = 90 b.) TYW and TYU are supplementary c.) VYW and TYS are complementary 25

26 Section 1 5: Angle Relationships Extra Examples Date: Example #1: Two angles are complementary. One angle measures 24 more than the other. Find the measures of the angles. Example #2: Find the measures of two supplementary angles if the measure of one angle is 4 less than 3 times the measure of the other angle. Example #3: The measure of an angle s supplement is 22 less than the measure of the angle. Find the measure of the angle and its supplement. Example #4: Find the value of x so that suur AC and suur BD are perpendicular. 26

27 CRITICAL THINKING 1.) A counterexample is used to show that a statement is not necessarily true. Find a counterexample for the statement Supplementary angles for linear pairs. 2.) What kinds of angles are formed when streets intersect? Include the following in your answer: the types of angles that might be formed by two intersecting lines, and a sketch of intersecting streets with angle measures and angle pairs identified. 27

28 Date: Section 1 6: Polygons Notes Polygons A polygon is a figure whose sides are all segments. The sides of each angle in a polygon are called of the polygon, and the vertex of each angle is a of the polygon. Examples: Polygons can be or. Examples: 28

Number of Sides 3 5 6 9 12 n Polygon quadrilateral heptagon octagon decagon Regular Polygon: a convex

Ex: Example #1: Name each polygon by the number of sides.

29 Number of Sides n Polygon quadrilateral heptagon octagon decagon Regular Polygon: a convex polygon in which all the are congruent and all the angles are. Ex: Example #1: Name each polygon by the number of sides. Then classify it as convex or concave, regular or irregular. a.) b.) Perimeter The perimeter of a polygon is the sum of the of its sides, which are. Example #2: Find the perimeter of each polygon. a.) b.) c.) 29

CRITICAL THINKING 1.) Refer to the figure to the right. Find the perimeter of pentagon LMNOP. Suppose the length of each side of pentagon LMNOP is doubled.

30 CRITICAL THINKING 1.) Refer to the figure to the right. Find the perimeter of pentagon LMNOP. Suppose the length of each side of pentagon LMNOP is doubled. What effect does this have on the perimeter? 2.) Quadrilateral ABCD has a perimeter of 95 centimeters. Find the length of each side if AB = 3a + 2, BC = 2(a 1), CD = 6a + 4, and AD = 5a 5. 30

Section 1-1 Points, Lines, and Planes

Section 1-1 Points, Lines, and Planes I CAN. Identify and model points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in space. Undefined Term- Words, usually