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The Geometry of Lines and Angles Connections Have you ever . . . • Looked at the patterns of streets on a map? • Marked out the location of a fence? • Measured parallel lines to hang two pictures on the wall? Lines and angles are all around us in everyday shapes. To understand distances, shapes, and forms, geometry looks at lines and the angles they form wherever they meet. A line in geometry is a two-dimensional figure on a plane. Lines can be parallel (;) or intersecting. Parallel lines never meet. Intersecting lines meet like two streets at an intersection. • Perpendicular (=) intersecting lines form four right angles. Intersecting Lines Perpendicular b c a 1 2 4 3 d • Intersecting lines form congruent (equal sized) opposite angles: E1 , E3 and E2 , E4 • The sum of two angles on a line is 180°: E1 + E2 =180c Transversal Lines Perpendicular p m n t r 5 6 7 8 9 10 s 11 12 A line crossing two parallel lines is called a transversal. • Perpendicular (=) transversals form eight right angles. • Transversals form two congruent sets of angles: E5 , E8 , E9 , E12 and E6 , E7 , E10 , E11 • If you draw a transversal at an angle, you can see which angles appear congruent. 203 Essential Math Skills Learn It! Intersecting Lines: Find All the Angles Use your knowledge of intersecting lines to deduce unknown information. • Find angles that you know. • Describe other angles based on their relationship to known angles. The diagram shows the intersections of four streets. 10 = 40° u t 2 1 • A Street (line a) ; B Street (line b). 4 3 • Tarry Lane (line t) and Uptown Drive (line u) form transversals. a 5 6 7 8 10 11 9 • Uptown Drive is = to both A Street and B Street. b 12 14 13 An architect needs to know the measure of E1 to plan a building on that lot. Mark Angles You Know Find all the angles that you know. Remember, perpendicular (=) lines give you information about angles. Mark angles that are given and angles formed by perpendicular lines. ? 1. What are the measurements of all the angles that you know? Mark them on the diagram. You are given the measure of one angle (E10 = 40c). Because Uptown Drive is perpendicular to two parallel streets, you know it forms right angles: E5 = 90c E7 = 90c E11 = 90c E14 = 90c E6 = 90c E8 = 90c E12/13 = 90c E9/10 = 90c 10 = 40° u t 2 1 3 4 90˚ 90˚ 90˚ 90˚ 90˚ 10 90˚ 9 90˚ b 12 13 Mark Angles You Can Find by Subtraction Mark all the angles that you can find by subtracting a portion of a bigger angle. 204 a 90˚ The Geometry of Lines and Angles ? 2. What are the measurements of angles you can find by subtraction? Mark them on the diagram. Since E9/10 is a right angle, you can find E9 by subtraction. 90c - 40c = 50c Mark Opposite Angles When two lines intersect, opposite angles (angles across from each other) are congruent. Identify and mark the measurements of opposite angles. ? 3. What are the measurements of all opposite angles? Mark them on the diagram. You can mark the angles opposite to E9 and E10: E13 = 40c and E12 = 50c Mark Congruent Angles on a Parallel Line When a transversal crosses two parallel lines, it forms two congruent sets of angles. For example, E1, E2, E3, and E4 are congruent to E9, E10/11, E12, and E13/14. ? 4. What are the measurements of E1, E2, E3, and E4? Mark them on the diagram. Using Un P A C E1 and E4 are congruent, and they are the same as E9 and E12. E2 and E3 are congruent, and they are the same as E10/11 and E13/14. • E1 = 50c • E2 = 40c + 90c = 130c • E3 = 50c • E4 = 40c + 90c = 130c To understand a problem with lines, look for lines that cross each other, forming angles. Identify opposite angles, perpendicular lines, and parallel lines. Find the Solution derstand Identify what you need to know to solve the problem. In this case, the architect wants to know the measurement of E1. lan The answer is 50°. ttack heck 205 Essential Math Skills e ic Pract It! Solve each problem using relationships of lines and angles. 1. Scaffolds like this one are used for construction work. A scaffold is created using intersecting bars for support. An engineer is working on a design that has bars meeting at 65° angles. Find the value of EA, EB, and EC in the engineer's diagram. B C 65° A a. What are the measurements of angles you can find by subtraction? b. What are the measurements of opposite angles? 2. When homes are built, the contractor creates a frame for each room using parallel wood beams and supports that form transversals. An electrician wants to know the measurement of EX in the diagram so he can plan the wiring. 206 a. Mark angles you can find by subtraction, opposite angles, and congruent angles on the parallel lines. Explain how you know the measurements of these angles. b. What is the value of EX? X 135° The Geometry of Lines and Angles 3. Delores is an engineer who works with the local railway system. She is designing a transition track for trains to move onto when they are out of service. The transition track will veer off at a 50° angle. When the train returns to service going southbound on this track, at what angle will the train’s turn be? Explain your reasoning. 50° X 4. This map shows a bike path that crosses a road running parallel to a river. If EB is 77.2°, what is EH? Explain your reasoning. A B C D E F G H Review & Practice Get more practice with lines and angles on pages 335–336. 207 Essential Math Skills 5. Using your knowledge that two angles on a straight line equal 180°, explain why EA = EC in this diagram. A B D C 6. The sum of the angles of a triangle equal 180°. Based on this information and what you know about intersecting lines, mark the measures of all 12 angles in these three intersections. Explain how you calculated the measurements. Build Your Math Skills 61.5° 112° Extend rays, the lines that form the sides of an angle, out beyond the vertex, the point where the lines meet, to form intersecting lines. ray vertex 208 The Geometry of Lines and Angles Check Your Skills Use your knowledge of lines and angles to answer the following questions. 1. This diagram shows studs in a wall frame. The carpenter wants to know if the studs are parallel. How could the carpenter do this? a. Check that EA and EI are congruent b. Check that EA, EE, and EJ are congruent A c. Check that EA, EH, and EL are congruent C d. Check that EA, ED, and EL are congruent 2. D a.128.7° b.137.7° c.142.7° d.148.7° K H G A builder is creating the stair railing shown in the diagram. What is the measure of EA? 3. F E B I J L A 42.3° Two streets cross at the angle shown in the diagram. If EA measures 77.1°, what does EB measure? a.77.1° b.99.9° c.101.1° d.102.9° 4. Tycho is planning to cut custom wood pieces to inlay in a tabletop. Each wood piece is shaped as a parallelogram, formed by two sets of parallel lines as shown in the diagram. Tycho needs to measure the angles to set the blades on his saw. If EA is 112°, what does EB measure? A B D A C B 209 Essential Math Skills 5. In her afternoon football game, Alexa runs across the 40 yard line at an angle of 134.2°. She turns left and crosses the 20 yard line at 138.5° as shown in the diagram. At what angle did she turn? a.86.7° b.87.3° c.89.8° d.92.2° 134.2° 138.5° X° Use this diagram to answer questions 6 through 8. A C E G B D F H 6. In this diagram of a transversal crossing two parallel lines, how many angle measurements do you need to know to find all eight angle measurements? 7. In this diagram of a transversal crossing two parallel lines, if EB is 31°, which of the following is true? a. EH = 31c b. EE = 149c c. EC = 149c d. EA = 31c 8. In this diagram of a transversal crossing two parallel lines, which set of angles have the same measures? 210 a. ED, EF, EA b. EB, EH, EG c. EC, EG, EF d. EF, EB, EA Remember the Concept • Opposite angles are congruent. • Two angles that form a straight line add up to 180°. • A transversal through two parallel lines creates two sets of congruent angles. Answers and Explanations By substitution, EC + 180c - EA = 180c. The Geometry of Lines and Angles page 203 Intersecting Lines: Find All the Angles Practice It! pages 206–208 EC - EA = 0c EC = EA 6. 50.5° 1a.EC = 180c - 65c = 115c 129.5° You can find this angle by subtraction because the sum of two angles that make a straight line is 180°. 1b.EB = 65c 129.5° 50.5° 112° 68° 112° 68° EA = 115c 2a.180c - 135c = 45c 118.5° 61.5° 61.5° 118.5° Transversals form congruent sets of angles across parallel lines. 45° 135° 135° 45° 45° 135° 135° 45° 45° 135° 135° 45° 2b.45° 3. 130° The railroad track forms a straight line. The transition track leaves at a 50° angle. The 50° angle plus the angle the train returns at will add up to 180°. 50c + X = 180c X = 180c - 50c = 130c 4. If EB is 77.2°, EH is 102.8°. EB and ED form a straight line, so they add up to 180°. 180c - 77.2c = 102.8c Since EH is in the same position in a group of angles formed by the transversal crossing a parallel line, EH is congruent to ED. Therefore, EH is 102.8°. 5. Since two angles that form a straight line must equal 180°, EA + EB = 180c. For the same reason, EC + EB = 180c. Since EA + EB = 180c, EB = 180c - EA. The angles adjacent to the angles marked 112c and 61.5c form straight lines with those angles, so subtract the angle from 180c to find the adjacent angles. Angles adjacent to the 112c angle are 68c, and angles adjacent to the 61.5c angle are 118.5c. The opposite angles from the angles marked 112c and 61.5c measure the same as those angles. This gives you two angles of the triangle in the center. Since the sum of the three angles of the triangle will be 180c: 68c + 61.5c + x = 180c 129.5c + x = 180c x = 50.5c The angles adjacent to the 50.5c angle form straight lines with that angle, so they measure 129.5c. The angle opposite to the 50.5c angle is congruent to that angle, so it is 50.5c. Check Your Skills pages 209–210 1. c. Check that EA, EH, and EL are congruent 2. b. 137.7° 3. d. 102.9° 4. 68° If EA is 112°, EB is 180° minus 112°, or 68°. 5. b. 87.3° i Essential Math Skills By extending the two lines that show where Alexa runs, you get two transversals that intersect and form a triangle with the 20 yard line. You can use your knowledge of transversals and triangles to find the angles. 134.2° 138.5° 87.3° 92.7° 134.2° 41.5° 45.8° 6. One You can find every angle from only one angle. All angles can be calculated by subtraction, by identifying opposite angles, and by identifying angles formed by the transversal intersecting the parallel line. 7. b. EE = 149c 8. c. EC, EG, EF ii