Finding Measures of Angles Formed by Transversals Intersecting Parallel Lines
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1 Lesson 22 Finding Measures of Angles Formed by Transversals Intersecting Parallel Lines 8.G.5 1 Getting the idea The figure below shows two parallel lines, j and k. The parallel lines,, are intersected by a transversal t (a line that intersects two or more lines). Special pairs of angles are formed when a transversal intersects two lines. t j k Corresponding Angles Corresponding angles are on the same side of the transversal and on the same side of the parallel lines. Corresponding angles have the same measure. /1 is in the upper left section of the intersection of lines j and t. /5 is also in the upper left section of the intersection of lines k and t, so /1 and /5 are corresponding angles. /2 and /6, /3 and /7, and /4 and /8 are also pairs of corresponding angles. Alternate Exterior Angles Alternate exterior angles are outside the parallel lines and on opposite sides of the transversal. Alternate exterior angles have the same measure. /2 is above line j and /8 is below line k. They are on opposite sides of the transversal, so /2 and /8 are alternate exterior angles. /1 and /7 are also alternate exterior angles. Lesson 22: Finding Measures of Angles Formed by Transversals Intersecting Parallel Lines 229
2 Alternate Interior Angles Alternate interior angles are on the inside of the parallel lines and are on opposite sides of the transversal. Alternate interior angles have the same measure. /3 is below line j and /5 is above line k. The angles are on opposite sides of the transversal, so /3 and /5 are alternate interior angles. /4 and /6 are also alternate interior angles. Vertical Angles Vertical angles are opposite angles formed by two intersecting lines. Vertical angles have the same measure. /2 and /4 are both formed by the intersection of line j and line t. The angles are opposite of each other, so /2 and /4 are vertical angles. /1 and /3, /6 and /8, and /5 and /7 are also vertical angles. Same-Side Interior Angles Same-side interior angles are between parallel lines and on the same side of the transversal. Same-side interior angles are supplementary angles. The measures of supplementary angles add to 180. /3 and /6 are between the lines and to the right of the transversal, so /3 and /6 are same-side interior angles. /4 and /5 are also same-side interior angles. 230 Domain 4: Geometry
3 Example 1 Line m n, and intersected by transversal l. Identify all pairs of corresponding angles, alternate exterior angles, vertical angles, alternate interior angles, and same-side interior angles in the figure below. l m n Strategy Use the definitions to find all of the pairs of angles. Step 1 Find all of the pairs of corresponding angles. /8 and /3, /7 and /4, /6 and /1, and /5 and /2 are corresponding angles. Step 2 Find all of the pairs of alternate exterior angles. /8, /7, /2, and /1 are outside the two lines. /7 and /1 are on opposite sides of the transversal. /8 and /2 are on opposite sides of the transversal. Step 3 Find all of the pairs of vertical angles. /1, /2, /3, and /4 are formed by the intersection of lines n and l. /1 and /4, and /2 and /3 are vertical angles. /5, /6, /7, and /8 are formed by the intersection of lines m and l. /6 and /7, and /5 and /8 are vertical angles. Step 4 Find all of the pairs of alternate interior angles. /3, /4, /5, and /6 are between the lines. /4 and /6 are on opposite sides of the transversal. /3 and /5 are on opposite sides of the transversal. Step 5 Solution Find all of the pairs of same-side interior angles. /3, /4, /5, and /6 are between the lines. /3 and /6 are on the same side of the transversal. /4 and /5 are on the same side of the transversal. The pairs of angles can be identified as follows: Corresponding angles: /2 and /5; /1 and /6; /4 and /7; /3 and /8 Alternate exterior angles: /1 and /7; /2 and /8 Vertical angles: /1 and /4; /2 and /3; /6 and /7; /5 and /8 Alternate interior angles: /4 and /6; /3 and /5 Same-side interior angles: /3 and /6; /4 and /5 Lesson 22: Finding Measures of Angles Formed by Transversals Intersecting Parallel Lines 231
4 Example 2 In the figure below, a b, and both lines are cut by transversals c and d. What are the missing angle measures in the figure? a b c 40 d Strategy Break the figure into sections based upon transversals. Step 1 Focus on the angles formed by parallel lines a and b that are cut by transversal line c. a 3 50 b Create a table to help organize your work. c Angle Pair /3 and the angle marked 50 /3 and /7 Classification How Are the Measures Related When Lines Are Parallel? What Can I Figure Out? vertical angles equal measure m/ alternate interior angles equal measure m/ /3 and /10 corresponding angles equal measure m/ /3 and /6 same-side interior angles sum of 180 m/3 1 m/ m/ m/ /6 and /11 vertical angles equal measure m/ Domain 4: Geometry
5 Step 2 Focus on the angles formed by parallel lines a and b, cut by transversal line d. a 1 5 b d Angle Pair Classification How Are the Measures Related When Lines Are Parallel? What Can I Figure Out? /8 and the angle marked 40 vertical angles equal measure m/ /5 and /8 alternate interior angles equal measure m/ /1 and /8 corresponding angles equal measure m/ /9 and the angle marked 40 supplementary angles sum of 180 m/ m/ /9 and /12 vertical angles equal measure m/ Lesson 22: Finding Measures of Angles Formed by Transversals Intersecting Parallel Lines 233
6 Step 3 Fill in the known angle measures in the figure. Use this information to find the measures of the remaining angles. The three angles below line a form a straight line, so they have a sum of m/ m/ m/ a b c d /2 and /4 are vertical angles, so m/ Solution m/1 5 m/5 5 m/ m/2 5 m/ m/3 5 m/7 5 m/ m/6 5 m/ m/9 5 m/ Domain 4: Geometry
7 2 COACHED EXAMPLE In the figure below, lines g and h are parallel and cut by transversal m. What is the measure of /2? g h m The angle marked 150 and m/1 have a sum of since they form a Therefore, m/1 5, so m/1 5. Since line g and line h are, /1 and /2 are angles and the measures of /1 and /2 are,. m/2 5 Lesson 22: Finding Measures of Angles Formed by Transversals Intersecting Parallel Lines 235
8 3 LESSON PRACTICE Use the figure below for questions 1 and 2. Lines m and n are parallel and are cut by transversal l. l m n 1 List four different pairs of vertical angles. 2 Look at each pair of angles. Do the angles have the same measure? Select Yes or No. A. /3 and /2 Yes No B. /5 and /1 Yes No C. /4 and /6 Yes No D. /4 and /8 Yes No E. /3 and /6 Yes No 3 Draw a pair of parallel lines intersected by a transversal so at least one angle is a 45 angle. Label all of the angle measures in your drawing. 236 Domain 4: Geometry
9 Use the figure below for questions 4 and 5. a b c d 4 Classify each pair of angles. Write the angle pair in the correct box. /8 and /9 /4 and /9 /14 and /1 /12 and /13 /6 and /11 /8 and /4 Vertical Angles Corresponding Angles Alternate Interior Angles Alternate Exterior Angles 5 Aarav measured /9 using a protractor and found it to be 39. He said that /13 must also be 39 since /9 and /13 are corresponding angles. Do you agree with Aarav? Explain. Lesson 22: Finding Measures of Angles Formed by Transversals Intersecting Parallel Lines 237
10 6 Can a pair of angles be both vertical angles and corresponding angles at the same time? Use words, numbers, or a drawing to justify your answer. 7 In the figure below, j k, and both lines are cut by transversal t. Suppose line t was moved so that /2 had a greater measure. t j k How will the other angle measures change? Select True or False for each statement. 8 A. The measure of /1 will stay the same. True False B. The measure of /6 will increase. True False C. The measure of /7 will decrease. True False D. The measure of /4 will decrease. True False E. The measure of /5 will decrease. True False Santiago notices that when a pair of parallel lines is cut by a transversal and one angle is a right angle, all of the other angles are also right angles. Use words, numbers, or drawings to explain why Santiago s observation is true. 238 Domain 4: Geometry
11 9 In the figure below, AC ED and /ABC and /ECD are right angles. E A B C D Part A Find a pair of corresponding angles, a pair of alternate interior angles, and a pair of vertical angles in the figure. Corresponding angles: Alternate interior angles: Vertical angles: Part B and and and Find two pairs of angles from nabc and necd that have the same measure. Explain how you found your answer. Lesson 22: Finding Measures of Angles Formed by Transversals Intersecting Parallel Lines 239
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