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Topic 1 Parallel Lines Unit 2 Topic 1
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Explore: Angles formed by Parallel Lines Take a few minutes to watch the following video. http://www.khanacademy.org/math/geometry/angles/ v/angles--part-3 http://www.khanacademy.org/math/geometry/angles/ v/angles--part-3 You can scan the QR code to the right if you have a SmartPhone. Simply download a scanner app like RedLaser. Note: Parallel lines are lines that never meet.
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Information Adjacent angles are pairs of angles that share a common vertex and common side. Two special types of adjacent angles are described below. Complementary angles are two angles with a sum of 90 °. Supplementary Angles are two angles with a sum of 180 °. Straight Line Angles are angles along a straight line with a sum of 180 °. Perpendicular lines cross each other at right angles. They can be marked with one right angle symbol at the intersection point of the lines. The two lines in the figure shown on the right are perpendicular.
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Example 1 Finding missing angles In each figure, find the specified angle. a)b) c)d) Try this on your own first!!!!
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Example 1: Solutions Finding missing angles In each figure, find the specified angle. a)b) c)d)
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Information Opposite angles are formed when two lines cross each other. The measures of opposite angles are equal. In the figure shown on the right, a = b and c = d. Parallel lines do not cross each other. They can be marked by matching arrowheads located along the line. A transversal is a line that crosses two or more parallel lines. In the figure shown on the right, lines and are parallel. Line is a transversal.
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Information Corresponding angels are equal. Alternate interior angles are equal. Same side interior angles add to 180 °. x x x x x x
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Example 2 Identifying Angles Formed by Two Parallel Lines Identify the following pairs of angles shown in the figure on the right. a) opposite angles b) alternate interior angles c) corresponding angles d) same side interior angles Try this on your own first!!!! 12 34 56 87
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Example 2: Solution Identifying Angles Formed by Two Parallel Lines Identify the following pairs of angles shown in the figure on the right. a) opposite angles b) alternate interior angles c) corresponding angles d) same side interior angles 12 34 56 87
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Example 3 Determining if Two Lines Are Parallel In each figure, determine if lines AB and CD are parallel with each other. Explain why or why not. a) b) c) Try this on your own first!!!!
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Example 3: Solution Determining if Two Lines Are Parallel a) b) c) Parallel because the alternate interior angles are equal. Not parallel because the same side interior angles do not sum to 180 °. Parallel because opposites angles are equal and then corresponding angles are equal.
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Example 4a Determining Missing Angles Determine the measures of the missing angles in each of the diagrams. Try this on your own first!!!! 67 A F B D E C G Angle Measure A = B = C = D = E = F = G =
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67 A F B D E C G Angle MeasureReason A = 113 ⁰ Angles that lie beside each other are supplementary. (180-67=113). B = 67 ⁰ Opposite angles are equal. C = 113 ⁰ Opposite angles are equal. D = 113 ⁰ Alternate interior angles are equal (C=D). E = 67 ⁰ Interior angles are supplementary (E=180-C). 180-113=67. F = 67 ⁰ Alternate exterior angles are equal. G = 113 ⁰ Corresponding angles are equal. (C=G) Example 4a: Solution 113 ⁰ 67 ⁰ 113 ⁰ 67 ⁰ 113 ⁰ Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure.
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Example 4b Try this on your own first!!!! 138 A B C D E F G Angle Measure A = B = C = D = E = F = G =
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Example 4b: Solution Example has a video solution. Click here! Note: The solution has also been provided on the following slide.
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Example 4b: Solution 138 A B C D E F G Angle MeasureReason A = 42 Corresponding angles are equal. ( A = E) B = 138 Straight angle. ( A + B = 180 ) C =138 Same side interior angles. ( C + E = 180 ) D = 42 Alternate interior angles are equal. ( D = E ) E = 42 Straight angle (*started here). ( E + 138 = 180 ) F = 138 Opposite angles are equal. ( F = 138 ) G = 42 Opposite angles are equal. ( G = E) 138 42 138 Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure.
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Example 4c Try this on your own first!!!! a b c d 110 Angle Measure a = b = c = d =
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Example 4c: Solution a b c d 110 Angle MeasureReason a = 110 Corresponding angles are equal. b = 110 Opposite angles are equal. c = 70 Interior angles are supplementary. d = 70 Alternate interior angles are equal. 110 70 Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure.
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Example 4d Try this on your own first!!!! Angle Measure w = x = y = w x y 120
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Example 4d: Solution Angle MeasureReason w = 120 Opposite angles are equal. x = 60 Interior angles are supplementary. y = 60 Corresponding angles are equal. w x y 120 60 Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure.
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Example 4e Try this on your own first!!!! Angle Measure a = b = c = d = e = f = 55 112 a e d f c b
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Example 4e: Solution Example has a video solution. Click here! Note: The solution has also been provided on the following slide.
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Example 4e: Solution 55 112 a e d f c b Angle MeasureReason a = 112 Corresponding angles are equal. ( a = 112 ) b = 55 Opposite angles are equal. ( b = f) c =68 Straight angle. ( c + 112 = 180 ) d = 55 Opposite angles are equal. ( d = 55 ) e = 112 Alternate interior angles are equal. ( e = 112 ) f = 55 Alternate interior angles are equal. ( f = d) Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure. 112 55 68
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Example 5 Proving equal angles in parallel lines Given that lines k and m are parallel, prove that Try this on your own first!!!! A C B k m StatementReason
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Example 5: Solution A C B k m StatementReason Given Corresponding angles are equal. Opposite angles are equal.
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Need to Know: When a transversal intersects parallel lines, several relationships are formed. Opposite angles are equal angles, directly across from each other. Corresponding angles are equal angles, where one interior angle and one exterior angle are on the same side of the transversal (F rule). Alternate interior angles are equal angles, where two interior angles are on opposite sides of the transversal (Z rule).
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Need to Know: Interior angles are supplementary angles, where the two interior angles are on the same side of the transversal. (C rule). Supplementary angles are angles that have a sum of 180 . Alternate exterior angles are equal angles, where the two exterior angles are on the opposite sides of the transversal. You’re ready! Try the homework from this section.
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