Lesson 6: Solve for Unknown Angles Angles and Lines at a Point

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1 : Solve for Unknown Angles Angles and Lines at a Point Student Outcomes Students review formerly learned geometry facts and practice citing the geometric justifications in anticipation of unknown angle proofs. Lesson Notes Lessons 1 through 5 serve as a foundation for the main subject of this module, which is congruence. By the end of the Unknown Angles lessons (Lessons 6 8), students will start to develop fluency in two areas: (1) solving for unknown angles in diagrams and (2) justifying each step or decision in the proof writing process of unknown angle solutions. The missing angle problems in this topic occur in many American geometry courses and play a central role in some Asian curricula. A missing angle problem asks students to use several geometric facts together to find angle measures in a diagram. While the simpler problems require good, purposeful recall and application of geometric facts, some problems are complex and may require ingenuity to solve. Unfortunately, many geometry courses offered in the USA do not expect even this level of sophistication. They do not demand that students use their knowledge constructively but instead ask students to merely regurgitate information. The missing angle problems are a step up in problem solving. Why are they here at this juncture in this course? The main focuses of these opening lessons are to recall/refresh and supplement existing conceptual vocabulary, emphasize the idea that work in geometry involves reasoned explanations, and provide situations/settings that support the need for reasoned explanations and illustrate the satisfaction of building such arguments. The Problem Set assignment from Lesson 5 is a chart of all the geometry facts on angles covered before Grade 10. is a problem set based lesson and focuses on just a portion of this chart, solving for unknown angles in diagrams of angles and lines at a point. By the next lesson, students should be comfortable solving for unknown angles numerically or algebraically in diagrams involving supplementary angles, complementary angles, vertical angles, and adjacent angles at a point. As always, vocabulary is critical, and students should be able to define the relevant terms themselves. It may be useful to draw or discuss counterexamples of a few terms and elicit why they do not fit a particular definition. As students work on problems, encourage them to show each step of their work and to list the geometric reason for each step, such as vert. s. This will prime students for writing a reason for each step of their unknown angle proofs in a few days. Date: 4/8/14 44

2 Classwork Opening Exercise (5 minutes) Review the Problem Set from Lesson 5. Make sure students have precise answers in the chart. This chart is a guide for subsequent lessons. Discussion (2 minutes) Discussion MP.6 Two angles and, with a common side, are adjacent angles if belongs to the interior of. The sum of angles on a straight line is 180. Two angles are called supplementary if the sum of their measures is 180 ; two angles are called complementary if the sum of their measures is 90. Describing angles as supplementary or complementary refers only to the measures of their angles; the positions of the angles or whether the pair of angles is adjacent to each other is not part of the definition. In the figure, AB is a straight line. Find the measure of f. The total measure of adjacent angles around a point is 360. Find the measure of g. Vertical angles have equal measure. Find the measure of h. Date: 4/8/14 45

3 Example 1 (30 minutes) Example 1 Find the measure of each labeled angle. Give a reason for your solution. Angle Angle measure Reason a 35 s on a line b 140 s on a line c 40 vert. s d 140 s on a line or vert. s e 238 s at a pt. Students have a complete list of abbreviated reasons after the review of the Lesson 5 Problem Set. Exercises Exercises MP.7 In the figures below,,, and are straight lines. Find the measure of each marked angle or find the unknown numbers labeled by the variables in the diagrams. Give reasons for your calculations. Show all the steps to your solutions. 1., s on a line 2., s on a line Date: 4/8/14 46

4 3., s on a line 4., s on a line 5., s at a point 6., s at a point 7. ;,, vert. s and s at a point 8., vert. s and s on a line Date: 4/8/14 47

5 9. x x + 6 y x - 11 ;, vert. s and s at a point 10. y 2 5 x ;, s on a line 3 4 x 2 For problems 11 and 12, find the values of and. Show all the steps to your solution. 11. ; MP x + 1 y 3 y 3x x + 10 ; Relevant Vocabulary Relevant Vocabulary Straight Angle: If two rays with the same vertex are distinct and collinear, then the rays form a line called a straight angle. Vertical Angles: Two angles are vertical angles (or vertically opposite angles) if their sides form two pairs of opposite rays. Exit Ticket (5 minutes) Date: 4/8/14 48

6 Name Date Exit Ticket Use the following diagram to answer the questions below: 1. a. Name an angle supplementary to and provide the reason for your calculation. b. Name an angle complementary to and provide the reason for your calculation. 2. If = 38, what is the measure of each of the following angles? Provide reasons for your calculations. a. b. c. Date: 4/8/14 49

7 Exit Ticket Sample Solutions Use the following diagram to answer the questions below: 1. a. Name an angle supplementary to and provide the reason for your calculation. or b. Name an angle complementary to and provide the reason for your calculation. 2. If = 38, what is the measure of each of the following angles? Provide reasons for your calculations. a. b. c. Problem Set Sample Solutions Find the value of and/or in each diagram below. Show all the steps to your solution. 1. Date: 4/8/14 50

8 2. 3. Date: 4/8/14 51