PLANE GEOMETRY SKILL BUILDER ELEVEN

Transcription

1 PLANE GEOMETRY SKILL BUILDER ELEVEN Lines, Segments, and Rays The following examples should help you distinguish between lines, segments, and rays. The three undefined terms in geometry are point, line, and plane. X represents point X It has no size. represents line YZ It extends without end in both directions. represents plane W It has a flat surface that extends indefinitely in all directions. Segment = A part of a line consisting of two endpoints and all the points between them. AD and DA are the same segment. But AD and DA are different rays (different endpoints). AD and DA are the same line. BA and BC are opposite rays (same endpoints). Measurement and Construction of Right, Acute, and Obtuse Angles Naming Angles An angle is formed by two rays having a common endpoint. This endpoint is called the vertex of the angle. Angles are measured in degrees. Segment AB or segment BA Ray = A part of a line consisting of one endpoint and extending without end in the other direction. Angles may be named in three different ways: (1) by the letter at its vertex ( A). (2) by three capital letters with the vertex letter in the center ( BAC or CAB). (3) by a lower case letter or a number placed inside the angle ( x). Ray CD (endpoint must be named first) or CD Collinear points are points that lie on the same line. Non-collinear points are points that do not all lie on the same line. NOTE: NOTE: Angle C cannot be the name for the angle because there are three angles that have the common vertex, C. They are angles BCE, ECD, and BCD. Angle BCE may also be named ECB or 1. Angles 1 and 2 are adjacent angles since they share a common vertex, C, and a common side, CE, between them. 123

2 Classifying Angles Angles are classified according to the number of degrees contained in the angle. Supplementary angles are two angles whose sum is 180. Type of Angle acute angle right angle obtuse angle straight angle Number of Degrees less than greater than 90 but less than Since PQR = 180, angles 3 and 4 are supplementary angles. This is also an example of a linear pair, adjacent angles such that two of the rays are opposite rays that form a linear pair. Example If LN NM, express the number of degrees in x in terms of y. Acute BAC Right YXZ Straight TRS Since LN MN, LNM measures 90. x + y = 90 y y x = 90 y (using the additive inverse) Obtuse EDF Complementary and Supplementary Angles Example If PQ is a straight line, express y in terms of x. Complementary angles are two angles whose sum is 90. A straight line forms a straight angle. Therefore Since ABC measures 90, angles 1 and 2 are complementary angles. y in terms of x x + y = 180 x x y = 180 x x in terms of y x + y = 180 y y x = 180 y (using the additive inverse) (using the additive inverse) 124

3 Example If BA AC, find the number of degrees in angle x. A Since BA AC, BAC = 90. Therefore, x + x + x + 45 = 90 3x + 45 = 90 (combining like terms) x = 45 (using the multiplicative 1 1 (3x ) = (45 ) 3 3 x = 15 C (using the additive inverse) inverse) Example Find the number of degrees in angle x. Angles 1 and 3 are vertical angles. Angles 2 and 4 are also vertical angles. If m 2 = 50, then m 1 + m 2 = 180 (straight AB) and 1 measures 130. Also, m 1 + m 2 = 180 (straight CD), and 3 measures 130. Since supplements of the same angle are equal, vertical angles contain the same number of degrees. Thus, 1 3 and 2 4. Example Find x, y, and z. Since the five angles center about a point, their sum is 360. Therefore, x + x + x + x + x = 360 (combining like terms) 5x = (5x) = (360 ) 1 (using the multiplicative 5 5 Vertical Angles x = 72 inverse) Vertical angles are the non-adjacent angles formed when two straight lines intersect. Since vertical angles are equal, y = 105. The same is true for x and z: x = z. Any two adjacent angles such as z and 105 are supplementary. Therefore, z = (using the additive inverse) z = 75 x = 75 and y = 105 Perpendicular Lines Perpendicular lines are lines that meet and form right angles. The symbol for perpendicular is. 125

4 The arrows in the diagram indicate that the lines are parallel. The symbol means is parallel to : AB CD. AB is perpendicular to CD or AB CD If two intersecting lines form adjacent angles whose measures are equal, the lines are perpendicular. There are relationships between pairs of angles with which you are familiar from your previous studies. Angles 1 and 4 are vertical angles and congruent. Angles 5 and 7 are supplementary angles, and their measures add up to 180. Corresponding angles are two angles that lie in corresponding positions in relation to the parallel lines and the transversal. For example, 1 and 5 are corresponding angles. So are 4 and 8. Other pairs of corresponding angles are 2 and 6, 3 and 7. Angles 3 and 5 are interior angles on the same side of the transversal. These angles are supplementary. Angles 4 and 6 are also supplementary. If m 1 = m 2, then AB XY Perpendicular lines form four right angles. Parallel Lines and Transversals Angles Formed by Parallel Lines The figure illustrates two parallel lines, AB and CD, and an intersecting line EF, called the transversal. Alternate interior angles are two angles that lie on opposite (alternate) sides of the transversal and between the parallel lines. For example, 3 and 6 are alternate interior angles, as are 4 and 5. If two lines are parallel, corresponding angles are congruent or equal, and alternate interior angles are congruent or equal. In the diagram below, if l m, the alternate interior angles are congruent, and x z. Since corresponding angles are congruent, y z. Therefore, x y z. 126

5 Example l m and m a = 100. Find the number of degrees in angles b, c, d, e, f, g, and h. Example AB ED, B = 70 and ACB = 65. Find the number of degrees in x. Because a and d are vertical angles, d measures 100. Using supplementary angles, m b = = 80 and m c = = 80 ; therefore m b = m c = 80. Use either property of parallel lines corresponding angles or alternate interior angles to obtain the remainder of the answers. For example, m b = m f by corresponding angles or m c = m f by alternate interior angles. Thus, m b = m c = m g = m f = 80 and m e = m h = m d = 100. Knowing two angles of ABC, A = 45. Angles A and E are alternate interior angles. Therefore, A = E, since AB DE. Thus x =

6 Orientation Exercises 1. Which rays form the sides of ABC? A. AB, AC D. BA, BC B. AB, CB E. None of the above C. AC, BD 7. In the figure below, parallel lines AC and BD intersect transversal MN at points x and y. MXA and MYB are known as: 2. In the figure below, line a is: A. a bisector D. perpendicular B. parallel E. an altitude C. a transversal 3. Which angles appear to be obtuse? A. 2 and 4 D. 1 and 5 only B. 2, 3, and 4 E. 3 only C. 1, 3, and 5 A. vertical angles B. alternate interior angles C. complementary angles D. supplementary angles E. corresponding angles 8. In the figure below, AB CD and RS and PQ are straight lines. Which of the following is true? 4. Which angles form a pair of vertical angles? A. 1 and 2 D. 4 and 1 B. 2 and 4 E. 1 and 3 C. 3 and 4 5. At how many points will two lines that are perpendicular intersect? A. 0 D. 3 B. 1 E. 4 C If two intersecting lines form congruent adjacent angles, the lines are: A. parallel D. vertical B. oblique E. perpendicular C. horizontal A. g = z D. g = x B. g = y E. g = e C. g = f 9. The sum of the interior angles of a pentagon is: A. 480 D. 720 B. 540 E. 960 C If the perimeter of a square is 24x, its area is: A. 81x D. 48x 2 B. 36x 2 E. 81x 2 C. 24x 128

7 Practice Exercise Three points, R, S, and T are collinear. Point S lies between R and T. If RS = 3 2 RT and RS = 48, find 2 1 RT. A. 72 D. 36 B. 60 E. 24 C Points E, F, and G are collinear. If EF = 8 and EG = 12, which point cannot lie between the other two? A. E D. F and G B. F E. Cannot be determined C. G 3. If PRQ is a straight line, find the number of degrees in w. 5. Line XY is perpendicular to line CD at D. Which conclusion can be drawn? A. XD = DY B. XY = CD C. m XDC = 90 D. m XDC = 90 and XD = DY E. All of the above 6. In the figure, a, b, and c are lines with a b. Which angles are congruent? A. 4, 5 D. 2, 5 B. 4, 6 E. 2, 6 C. 4, 3 7. l m, and AB = AC. Find x. A. 30 D. 70 B. 50 E. 100 C In the figure, if AB is a straight line and m CDB = 60, what is the measure of CDA? A. 40 D. 100 B. 60 E. None of the above C In the figure, if lines r and s are parallel, what is the value of x? A. 15 D. 90 B. 30 E. 120 C. 60 A. 30 D. 120 B. 60 E. 150 C

8 9. The height of the triangle below is 10 units. What is its area? 10. The measure of the smaller angle in figure below is: 15x x 5 A. 150 B. 300 C. 340 D. 600 E. 680 A. 55 B. 75 C. 105 D. 125 E

9 Practice Test In the figure, U, V, W, and X are collinear. UX is 50 units long, UW is 22 units long, and VX is 29 units long. How many units long is VW? A. 1 D. 21 B. 7 E. 28 C P, Q, R, S, and T are five distinct lines in a plane. If P Q, Q R, S T, and R S, all of the following are true, except: A. P R D. S Q B. P S E. Q T C. P T 6. In the figure, if l 1 l 2, l 2 l 3, and l 1 l 4, which of the following statements must be true? 2. How many different rays can be named by three different collinear points? A. 0 D. 3 B. 1 E. 4 C A 60 angle is bisected, and each of the resulting angles is trisected. Which of the following could not be the degree measure of an angle formed by any two of the rays? A. 10 D. 40 B. 20 E. 50 C Solve for x. A. 45 D. 16 B. 40 E. None of the above C. 20 I. l 1 l 3 II. l 2 l 4 III. l 3 l 4 A. None D. II and III only B. I only E. I, II, and III C. I and II only 7. When two parallel lines are cut by a transversal, how many pairs of corresponding angles are formed? A. 1 D. 4 B. 2 E. 8 C If a b c and c d, which of the following statements is true? A. a d D. b c B. a c E. a d C. b d 131

10 9. The perimeter of the triangle below is: 10. A letter carrier must go from point A to point B through point C in order to make his delivery. How much distance could he save if he could go directly from point A to point B and not pass through point C? A. 54 B. 66 C. 42 D. 74 E. 40 A. 750 ft B. 200 ft C. 500 ft D. 350 ft E. 600 ft 132