Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Technical drawing wikipedia , lookup
History of geometry wikipedia , lookup
Rotation formalisms in three dimensions wikipedia , lookup
Integer triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Line (geometry) wikipedia , lookup
Perceived visual angle wikipedia , lookup
Multilateration wikipedia , lookup
Trigonometric functions wikipedia , lookup
M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES ASSIGNMENTS FOR PART 1 OF UNIT 2 LINES AND ANGLES Part 1 of Unit 2 includes sections 1-4, 1-5, and 2-8 from our textbook. Due Number 2A 2B 2C 2D Description Topics Section 1-4: Vocabulary: degrees, acute angle, obtuse angle, right angle, angle bisector Name and label angles using a number, single p. 41-42 # 9, 10, 12, 18, 20 – 23 all, 43 letter, or three letters p. 44 # 56 Identify sides and vertex of an angle Use a protractor to measure angles in degrees Classify an angle as right, acute, or obtuse Use algebra to solve problems involving an angle bisector Section 1-5: Vocabulary: adjacent angles, linear pair, vertical angles, complementary, p. 51-52 # 8 – 17 all, 36 – 41 all supplementary, perpendicular Identify pairs of angles that have specific relationships Section 1-5: p. 51 # 19 – 26 all Use algebra to solve problems involving angle pairs Section 2-8: p. 156 – 157 # 1 – 4 all, 6, 8 – 10 all, 13 Use two-column proofs to prove statements about angle pairs. 1 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 1-4 1-4 Angle Measure Part 1: Naming Angles Figure 1 If two rays have a common endpoint, they form an angle. The rays are the sides of the angle. The common endpoint is the vertex. 1. Name the sides of the angle shown in Figure 1. 2. Name the vertex of the angle shown in Figure 1. 3. The angle in Figure 1 can be named as ∠ A, ∠ BAC, ∠ CAB, or ∠ 1. Use this information to write three different rules for naming angles: a. __________________________________________________________________________ b. __________________________________________________________________________ c. __________________________________________________________________________ 4. Use Figure 2 to answer the questions below: a. Name all the different angles that have R as their vertex. Avoid naming the same angle more than once. Figure 2 b. Name the sides of ∠ 1. A right angle is an angle whose measure is 90 ° . An acute angle has measure less than 90 ° . An obtuse angle has measure greater than 90 ° but less than 180 ° . 2 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 5. Classify each angle in Figure 3 as right, acute, or obtuse. Then use a protractor to measure the angle to the nearest degree. a. ∠ ABD b. ∠ DBC c. ∠ EBC Figure 3 Part 2: Congruent Angles Figure 4 Angles that have the same measure are congruent angles. A ray that divides an angle into two congruent angles is is the angle bisector called an angle bisector. In Figure 4, of ∠ MPR, and ∠ MPN ≅ ∠ NPR. Point N is in the interior of ∠ MPR. In the diagram, arcs are drawn inside these angles to show that they are congruent. 6. Follow the steps below to solve the problem: In Figure 4, if m ∠ MPN = 2x + 14 and m ∠ NPR = x + 34, find x, and find m ∠ NPR. a. Since is an angle bisector of ∠ MPR, then ∠ MPN ≅ ∠ NPR. Write an equation about x showing that the measures of these angles are equal. b. Solve your equation for x. c. Use the value of x from part (b) to find m ∠ NPR. 3 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 7. In Figure 5, and are opposite rays. (This means bisects ∠ PQT. that they lie on the same line.) Also, a. Write an equation, and solve: If m ∠ PQT = 60 and m ∠ PQS = 4x + 14, find the value of x. Figure 5 b. Write an equation, and solve: If m ∠ PQS = 3x + 13 and m ∠ SQT = 6x – 2, find m ∠ PQT. c. Name a point in the interior of ∠ PQT. 4 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 1-5 Angle Relationships 1-5 Part 1: Pairs of Angles Adjacent angles are two angles that lie in the same plane and have a common vertex and a common side, but no common interior points. 1. Name a pair of adjacent angles in the diagram at right. 2. Name a different pair of adjacent angles in the same diagram. 3. The chart below shows pairs of angles that are vertical angles on the left and pairs of angles that are not vertical angles on the right. Pairs of Vertical Angles NOT Pairs of Vertical Angles ∠1 and ∠2 ∠3 and ∠4 ∠AED and ∠BEC ∠AEC and ∠DEB ∠1 and ∠2 ∠3 and ∠4 ∠5 and ∠6 ∠7 and ∠8 ∠9 and ∠10 Use the examples above to write a definition for vertical angles. Use the words “sides” and “vertex” in your definition. Vertical angles are two angles that… 5 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 4. The chart below shows pairs of angles that form a linear pair on the left and pairs of angles that do not form a linear pair on the right. Linear Pairs NOT Linear Pairs ∠1 and ∠2 ∠3 and ∠4 ∠AED and ∠AEC ∠BED and ∠DEA ∠1 and ∠2 ∠3 and ∠4 ∠5 and ∠6 ∠A and ∠B Use the examples above to write a definition for a linear pair. Use the words “sides” and “vertex” in your definition. A linear pair is a pair of angles that… 5. The chart below shows pairs of angles that are complementary angles on the left and pairs of angles that are not complementary angles on the right. NOT Complementary Angles Complementary Angles m∠1 + m∠2 = 90° ∠G and ∠H ∠1 and ∠2 ∠3 and ∠4 ∠1 and ∠2 ∠3 and ∠4 Use the examples above to write a definition for complementary angles. Complementary angles are a two angles that… 6 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 6. The chart below shows pairs of angles that are supplementary angles on the left and pairs of angles that are not complementary angles on the right. Supplementary Angles m∠3 + m∠4 = 180° NOT Supplementary Angles m∠4 + m∠5 ≠ 180° ∠1 and ∠2 ∠3 and ∠4 ∠1, ∠2, and ∠3 ∠4 and ∠5 Use the examples above to write a definition for supplementary angles. Supplementary angles are a two angles that… Figure 6 7. Use the Figure 6 to answer the following questions: a. Name a pair of vertical angles. b. Name a pair of adjacent angles. c. Name a pair of supplementary angles. d. Name a pair of complementary angles. 7 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES Part 2: Algebra and Perpendicular Lines Lines, rays, and segments that form four right angles are perpendicular. The right angle symbol □ indicates that the lines are perpendicular. In the is perpendicular to , or ⊥ . figure at the right, 8. Follow the steps below to solve the problem: Find x so that are perpendicular. and a. If ⊥ , then m ∠ DZP = 90 ° . Write an equation showing that the measures of ∠ DZQ and ∠ QZP add up to 90° . b. Solve your equation for x. c. Check your answer by finding the measures of ∠DZQ and ∠QZP and adding them. ⊥ 9. In Figure 7, find the value of x and y so that 8 . Figure 7 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES Part 3: Properties of Angle Pairs 10. In Figure 8, C lies between points A and B on . a. Name the linear pair of angles in Figure 8. b. Find the sum of the measures of the linear pair in Figure 8. Figure 8 c. Use a straightedge to draw another linear pair of angles. d. Use a protractor to measure the two angles you drew, and find their sum. e. Complete this statement: If two angles form a linear pair, then their measures… 11. Use Figure 9 to answer the following: Figure 9 a. Find each measure: m∠DEA = ______ m∠AEC = ______ m∠CEB = ______ b. Name two pairs of vertical angles in Figure 9. What do you notice about their measures? 9 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 12. Use Figure 10 to answer the following: a. Find each measure: m∠HJG = ______ m∠IJF = ______ m∠GJI = ______ b. Name two pairs of vertical angles in Figure 10. What do you notice about their measures? c. Complete the statement: Vertical angles have measures that… 10 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 2-8 2-8 Proving Angle Relationships Angle Addition Two adjacent angles can be added to form a single angle with measure equal to the sum of their measures. Example: In the diagram at right, m ∠ PQR + m ∠ RQS = m ∠ PQS. Linear pairs add up to 180. If two angles form a linear pair, then they are supplementary. Example: If ∠ 1 and ∠ 2 form a linear pair, then m ∠ 1 + m ∠ 2 = 180. Two angles that form a right angle add up to 90. If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary. Example: If ⊥ , then m ∠ 3 + m ∠ 4 = 90. These examples show how your new rules can be used. Example 1: If ∠ 1 and ∠ 2 form a linear pair and m ∠ 2 = 115, find m ∠ 1. m ∠ 1 + m ∠ 2 = 180 m ∠ 1 + 115 = 180 m ∠ 1 = 65 Linear pairs add up to 180. Substitution Subtraction Property Example 2: If ∠1 and ∠2 form a right angle and m∠2 = 20, find m∠1. m ∠ 1 + m ∠ 2 = 90 m ∠ 1 + 20 = 90 m ∠ 1 = 70 Two angles that forma right angle add up to 90. Substitution Subtraction Property Write an equation and solve to find the measure of each numbered angle. Name the reason that justifies your work. 1. 2. m ∠ 7 = 5x + 5, m∠8 = x – 5 3. m ∠ 5 = 5x, m ∠ 6 = 4x + 6, m ∠ 7 = 10x, m ∠ 8 = 12x – 12 11 m ∠ 11 = 11x, m ∠ 13 = 10x + 12 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES The Reflexive, Symmetric, and Transitive Properties all hold true for angles. The following theorems also hold true for angles. Congruent Supplements Theorem Angles supplementary to the same angle or congruent angles are congruent. Congruent Complements Theorem Angles complimentary to the same angle or to congruent angles are congruent. Vertical Angles Theorem If two angles are vertical angles, then they are congruent. Theorem 2.9 Perpendicular lines intersect to form four right angles. Theorem 2.10 All right angles are congruent. Theorem 2.11 Perpendicular lines form congruent adjacent angles. Theorem 2.12 If two angles are congruent and supplementary, then each angle is a right angle. Theorem 2.13 If two congruent angles form a linear pair, then they are right angles. Example: Write a two-column proof. Given: ∠ ABC and ∠ CBD are complementary. ∠ DBE and ∠ CBD form a right angle. Prove: ∠ ABC ≅ ∠ DBE Statements Reasons 1. ∠ABC and ∠CBD are complementary. ∠DBE and ∠CBD form a right angle. 1. Given 2. ∠DBE and ∠CBD are complementary. 2. Complement Theorem 3. ∠ABC ≅ ∠DBE 3. s complementary to the same ∠ or ≅ Complete each proof. 4. Given: ⊥ ; ∠ 1 and ∠ 3 are complementary. Prove: ∠ 2 ≅ ∠ 3 Proof: Statements a. Reasons a. ______________ ⊥ b. _________________ b. Def. of ⊥ c. m ∠ ABC = 90 c. Def. of right angle d. m ∠ ABC =m ∠ 1 + m ∠ 2 d. ______________ e. 90 = m ∠ 1 + m ∠ 2 e. Substitution f. ∠ 1 and ∠ 2 are complementary g. _________________ f. _______________ g. Given h. ∠ 2 ≅ ∠ 3 h. ______________ 12 s are ≅. M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 5. Given: ∠ 1 and ∠ 2 form a linear pair. m ∠ 1 + m ∠ 3 = 180 Prove: ∠ 2 ≅ ∠ 3 Proof: Statements a. ∠ 1 and ∠ 2 form a linear pair. m ∠ 1 + m ∠ 3 = 180 b. _________________ c. ∠ 1 is suppl. to ∠ 3. d. _________________ Reasons a. Given b. Suppl. Theorem c. ________________ d. ∠ s suppl. to the same ∠ or ≅ ∠ s are ≅. 13 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES PRACTICE FOR SECTIONS 1-4, 1-5, 2-8 For # 1-12, use the figure at the right. Name the vertex of each angle. 1. ∠ 4 2. ∠ 1 3. ∠ 2 4. ∠ 5 7. ∠ STV 8. ∠ 1 11. ∠ WTS 12. ∠ 2 Name the sides of each angle. 5. ∠ 4 6. ∠ 5 Write another name for each angle. 9. ∠ 3 10. ∠ 4 Classify each angle as right, acute, or obtuse. Then use a protractor to measure the angle to the nearest degree. 13. ∠ NMP 14. ∠ OMN 15. ∠ QMN 16. ∠ QMO 14 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES In the figure, and are opposite rays, bisects ∠ EBC. Write an equation, and solve to find the indicated measures. 17. If m ∠ EBD = 4x + 16 and m ∠ DBC = 6x + 4, find m ∠ EBD. 18. If m ∠ EBD = 4x − 8 and m ∠ EBC = 5x + 20, find the value of x and m ∠ EBC. For # 19- 24, use the figure at the right. Name an angle or angle pair that satisfies each condition. 19. Name two acute vertical angles. 20. Name two obtuse vertical angles. 21. Name a linear pair. 22. Name two acute adjacent angles. 23. Name an angle complementary to ∠ EKH. 24. Name an angle supplementary to ∠ FKG. 15 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES 25. Write an equation, and solve: Find the measures of an angle and its complement if one angle measures 24 degrees more than the other. 26. Write an equation, and solve: The measure of the supplement of an angle is 36 less than the measure of the angle. Find the measures of the angles.. For # 27-28, use the figure at the right. 27. If m ∠ RTS = 8x + 18, find the value of x so that ⊥ . 28. If m ∠ PTQ = 3y –10 and m ∠ QTR = y, find the value of y so that ∠ PTR is a right angle. Determine whether each statement can be assumed from the figure. Explain. 29. ∠ WZU is a right angle. 30. ∠ YZU and ∠ UZV are supplementary. 31. ∠ VZU is adjacent to ∠ YZX. 16 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES Find the measure of each numbered angle, and give the reason(s) that justify your work. 32. m ∠ 2 = 57 33. m ∠ 5 = 22 34. m ∠ 1 = 38 35. m ∠ 13 = 4x + 11, m ∠ 14 = 3x + 1 36. ∠ 9 and ∠ 10 are complementary. ∠ 7 ≅ ∠ 9, m ∠ 8 = 41 37. m ∠ 2 = 4x – 26, m ∠ 3 = 3x + 4 38. Complete the following proof. Given: ∠ QPS ≅ ∠ TPR Prove: ∠ QPR ≅ ∠ TPS Proof: Statements Reasons a.______________________________________ b. m ∠ QPS = m ∠ TPR c. m ∠ QPS = m ∠ QPR + m ∠ RPS m ∠ TPR = m ∠ TPS + m ∠ RPS d. ______________________________________ e. ______________________________________ f. ______________________________________ a. ________________________________________ b. ________________________________________ c. ________________________________________ 17 d. Substitution e. _______________________________________ f. _______________________________________ M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES REVIEW FOR SECTIONS 1-4, 1-5, 2-8 For # 1-10, use the figure at the right. Name the vertex of each angle. 1. ∠ 5 2. ∠ 3 3. ∠ 8 4. ∠ NMP 7. ∠ MOP 8. ∠ OMN Name the sides of each angle. 5. ∠ 6 6. ∠ 2 Write another name for each angle. 9. ∠ QPR 10. ∠ 1 Classify each angle as right, acute, or obtuse. Then use a protractor to measure the angle to the nearest degree. 11. ∠ UZW 12. ∠ YZW 13. ∠ TZW 14. ∠ UZT In the figure, and are opposite rays, bisects ∠ DCF, and 15. If m ∠ DCE = 4x + 15 and m ∠ ECF = 6x – 5, find m ∠ DCE. 16. If m ∠ FCG = 9x + 3 and m ∠ GCB = 13x – 9, find m ∠ GCB. 18 bisects ∠ FCB. M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES Name an angle or angle pair that satisfies each condition. 17. Name two obtuse vertical angles. 18. Name a linear pair with vertex B. 19. Name an angle not adjacent to, but complementary to ∠ FGC. 20. Name an angle adjacent and supplementary to ∠ DCB. 21. Write an equation, and solve: Two angles are complementary. The measure of one angle is 21 more than twice the measure of the other angle. Find the measures of the angles. 22. Write an equation, and solve: If a supplement of an angle has a measure 78 less than the measure of the angle, what are the measures of the angles? For # 23-24, use the figure at the right. 23. If m ∠ FGE = 5x + 10, find the value of x so that ⊥ . 24. If m ∠ BGC = 16x - 4 and m ∠ CGD = 2x + 13,find the value of x so that ∠ BGD is a right angle . 19 M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES Determine whether each statement can be assumed from the figure. Explain. 25. ∠ NQO and ∠ OQP are complementary. 26. ∠ SRQ and ∠ QRP is a linear pair. 27. ∠ MQN and ∠ MQR are vertical angles. Find the measure of each numbered angle, and give the reasons that justify your work. 28. m ∠ 1 = x + 10 m ∠ 2 = 3x + 18 29. m ∠ 4 = 2x – 5 m ∠ 5 = 4x – 13 20 30. m ∠ 6 = 7x – 24 m ∠ 7 = 5x + 14