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January 5, 2011 1) 2) 3) Write your homework in your agenda: Angle worksheet Let’s practice constructing parallel and perpendicular lines. Then, let’s discuss angles and their relationship to one another. Take out a piece of paper to work out a few problems. Parallel Lines and Transversals The angle relationships that are formed What am I learning today? Constructions What will I do to show that I learned it? Determine the measurements of any given angle created by the intersection of parallel lines by a transversal Parallel Lines and Transversals What would you call two lines which do not intersect? Parallel B A D C The symbol || is used to indicate parallel lines. AB || CD Parallel Lines and Transversals A slash through the parallel symbol || indicates the lines are not parallel. B AB || CD A D C Parallel Lines and Transversals Transversal A transversal is a line which intersects two or more lines in a plane. The intersected lines do not have to be parallel. j k m t Lines j, k, and m are intersected by line t. Therefore, line t is a transversal of lines j, k, and m. Parallel Lines and Transversals Identifying Angles - j 1 3 2 4 k 5 7 6 8 t Alternate interior angles ___________________ are on the interior of the two lines and on opposite sides of the transversal. Alternate interior angles are: 3 and or 6, 4 and 5 Parallel Lines and Transversals Identifying Angles - j 1 3 2 4 k 5 7 6 8 t Alternate exterior angles ___________________ are on the exterior of the two lines and on opposite sides of the transversal. Alternate exterior angles are: 1 and or 8, 2 and 7 Parallel Lines and Transversals Identifying Angles - j 1 3 2 4 k 5 7 6 8 t Corresponding angles _____________________ are on the corresponding side of the two lines and on the same side of the transversal. Corresponding angles are: 1 and 5, 3 and 7, 2 and 6, 4 and 8 Parallel Lines and Transversals Identifying Angles - j 1 3 2 4 Vertical angles _______________are pairs of opposite congruent angles formed by intersecting lines. k 5 7 6 8 t Vertical angles are: 1 and 4, 2 and 3, 5 and 8, 6 and 7 Parallel Lines and Transversals Identifying Angles - j 1 3 2 4 Adjacent angles _________________are in the same plane and share a common vertex and a common side. k 5 7 6 8 t Adjacent angles are: 1 and 3, 2 and 4, 5 and 7, 6 and 8 Parallel Lines and Transversals Identifying Angles – Check for Understanding Determine if the statement is true or false. If false, correct the statement. 1. Line r is a transversal of lines p and q. True – Line r intersects 4 3 both lines in a plane. 2 1 5 6 8 7 2. 2 and 10 are alternate interior angles. 9 10 False - The angles are 11 12 16 15 14 13 corresponding angles on transversal p. Parallel Lines and Transversals Identifying Angles – Check for Understanding Determine if the statement is true or false. If false, correct the statement. 3. 3 and 5 are alternate interior angles. 1 2 8 7 3 4 6 5 9 10 11 12 16 15 14 13 False – The angles are vertical angles created by the intersection of q and r. 4. 1 and 15 are alternate exterior angles. True - The angles are alternate exterior angles on transversal p. Parallel Lines and Transversals Identifying Angles – Check for Understanding Determine if the statement is true or false. If false, correct the statement. 5. 6 and 12 are alternate interior angles. 1 2 8 7 True – The angles are alternate interior angles on transversal q. 3 4 6 5 6. 9 10 11 12 16 15 14 13 11 and 12 are complementary interior angles. False – The angles are supplementary. Parallel Lines and Transversals Identifying Angles – Check for Understanding Determine if the statement is true or false. If false, correct the statement. 7. 3 and 4 are alternate exterior angles. 1 2 8 7 9 10 False – The angles are supplementary. 3 4 6 5 11 12 16 15 14 13 8. 16 and 14 are corresponding angles. True – The angles are corresponding on transversal s. EXAMPLE 1 Identify congruent angles The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning. SOLUTION By the Corresponding Angles Postulate, m5 = 120°. Using the Vertical Angles Congruence Theorem, m4 = 120°. Because 4 and 8 are corresponding angles, by the Corresponding Angles Postulate, you know that m 8 = 120°. EXAMPLE 2 Use properties of parallel lines ALGEBRA Find the value of x. SOLUTION By the Vertical Angles Congruence Theorem, m 4 = 115°. Lines a and b are parallel, so you can use the theorems about parallel lines. m 4 + (x+5)° = 180° Consecutive Interior Angles Theorem 115° + (x+5)° = 180° Substitute 115° for m x + 120 = 180 x = 60 4. Combine like terms. Subtract 120 from each side. GUIDED PRACTICE for Examples 1 and 2 Use the diagram at the right. 1. If m 1 = 105°, find m SOLUTION m 4 = 105° m 5 = 105° m 8 = 105° 4, m 5, and m 8. GUIDED PRACTICE for Examples 1 and 2 Use the diagram at the right. 2. If m 3 = 68° and m your steps. 8 = (2x + 4)°, what is the value of x? Show GUIDED PRACTICE for Examples 1 and 2 SOLUTION m 7+m 8 = 180° m 3= m 7 m 3 = 68° 68° + 2x + 4 = 72 + 2x = 180° 180° 2x = 108 x = 54 7 and 8 are supplementary. Corresponding Angles Substitute 68° for m 7 and (2x + 4)for m Combine like terms. Subtract 72 from each side. Divide each side by 2. 8. January 6, 2011 1) 2) 3) Write your homework in your agenda: Part Three: Problem Sets worksheet Take out your homework and leave it on your desk. Here’s your Warm-Up…. WARM-UP… Parallel Lines are cut by a transversal to create parking spaces. Two angle measures are given. Determine the 8 angle measures and label the diagram. 3x + 2 + 2x – 4 = 180 x = 36.4 111.2 68.8 111.2 2x - 4 68.8 degrees = 68.8 111.2 68.8 3x + 2 = 111.2 degrees What am I learning today? Triangle Proportionality Theorem What will I do to show that I learned it? Use proportionality theorems to determine segment length If a line parallel to one side of a Q triangle intersects the other two sides, then it divides the two side S proportionally. T R U RT RU If TU ║ QS, then TQ US Finding the length of a segment AB ║ ED, BD = 8, DC = 4, and AE = 12. What is the length of EC? C 4 D 8 B E 12 A Steps: DC EC = BD AE 4 = EC 8 12 4(12) = EC 8 6 = EC C 4 D 8 B So, the length of EC is 6. E 12 A QUESTION What is the Triangle Proportionality Theorem? If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Q T R S U RT RU , then TU ║ QS. If TQ US Determining Parallels Given the diagram, determine whether MN ║ GH. LM G 56 = MG 8 = 21 3 21 M LN NH 56 = 48 16 8 L N 48 16 H = 3 3 1 ≠ 3 1 MN is not parallel to GH. QUESTION What is the Converse of the Triangle Proportionality Theorem? If three parallel lines intersect two transversals, then they divide the transversals proportionally. If r ║ s and s║ t and l and m intersect, r, s, and t, then UW VX r U WY t s W Y m V X Z XZ Using Proportionality Theorems the diagram 1 2 3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU? In P S 1 9 11 Q T 2 15 R U 3 PQ ST QR TU 9 11 15 TU 9 ● TU = 15 ● 11 15(11) 55 1 TU or18 9 3 3 P S 1 9 11 Q T 2 15 R U 3 QUESTION What is the Proportionality Theorem for Parallel Lines? EXAMPLE 1 Find the length of a segment In the diagram, QS || UT , RS = 4, ST = 6, and QU = 9. What is the length of RQ ? SOLUTION RQ QU = RQ = 9 RQ = 6 RS ST Triangle Proportionality Theorem 4 6 Substitute. Multiply each side by 9 and simplify. GUIDED PRACTICE 1. Find the length of YZ . SOLUTION XW WV 44 35 = XY YZ Triangle Proportionality Theorem = 36 YZ Substitute. 315 11 Simplify YZ = ANSWER So length of YZ = 315 11 GUIDED PRACTICE 2. Determine whether PS || QR . SOLUTION PQ PN = 50 90 = RS SN 5 9 = 40 72 ANSWER So Because = PS is parallel to QR PQ PN = RS SN PS || QR , = 5 9