8.1.2 What is its measure?

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1 8.1.2 What is its measure? Interior Angles of a Polygon Objective: Students will learn how to find the sum of the interior angles of a polygon and will be able CCSS: G-GMD.1 Mathematical Practices: reason abstractly and quantitatively Core: 8 13 to 8 15 HW: 8 17 to 8 23 Materials: Handouts, 1/student: RP 8.1.2, Learning Log 8-14d Suggested Lesson: Intro: Point out that the class will be looking for ways to find the sum of the interior angles of a polygon minutes Use Think-Ink-Pair-Share 8-14 distribute the RP 8.1.2, also LL 8-14d Discuss problem If time allows, Hot Potato. Closure (10 minutes): Ask each team to post a solution to one of the parts of problem 8 16 so that stude

2 Intro: Students will be looking for ways to find the sum of the interior angles of a polygon and that many different strategies are possible. Encourage students to look for different ways because they may find a method that seems easier or more efficient than their first method. Since students know the sum of the angles of a triangle is always 180, students should think back to what they have learned about polygons so far to help them decide how to dissect polygons into triangles minutes Use Think-Ink-Pair-Share.» Based on their experience with hinged mirrors and with pinwheel triangles in 8.1.1, students should consider the possibility of dividing the pentagon in problem 8-13 into isosceles triangles.» Students can reason that there are five triangles, so the sum of all the angles is (5) (180º) = 900º. However, since the central angles are not part of the interior angles, then 360 should be subtracted, 540º.» This method can be generalized as follows: For a polygon with n sides, the interior angle sum is 180n 360º» Some students may realize that the pentagon could instead be dissected into triangles as shown in the diagram at right. In this strategy, students connect vertices to create the minimum number of triangles inside the polygon. Later, students should notice that the number of triangles is always two less than the number of sides of the polygon. Thus, the interior angle sum for a polygon with n sides is 180(n 2). After 10 minutes, pull the class together and ask students to share how they found the sum. Encourage students to show how they dissected the pentagon and found the sum. As students share their solution method, be sure they provide reasons justifying why their method works. After each presentation, ask the class, Would this strategy work for other polygons? and Does it matter if the polygon is regular or not? 8-14 Distribute the RP and have teams start problem. Teams should use one of the strategies presented for problem 8-13 in order to find the interior angle sums of other polygons. Parts (c) and (d) require students to generalize to find the interior sum of a 100-gon and an n-gon. Discuss problem 8-13 before moving teams on to the next problems Students prove that the sum of the angles of a pentagon must be Use Hot Potato. This problem asks students to apply their understanding and connect it with other knowledge, such as finding the base angles of an isosceles triangle and using the relationships created when two parallel lines are cut by a transversal. Note: Part (b) of asks students to find the individual angle of an equiangular hexagon, a preview of Closure (10 minutes): Ask each team to post a solution to one of the parts of problem 8-16 so that students can compare strategies and correct their own work. Problem 8-14 asks students to write a Learning Log entry to summarize how to find the sum of the interior angles of a polygon. You could have students do a Peer Edit for this entry.

3 8-6 a. 110 b. 70 c. 48 d a. no b. yes c. no d. yes e. no f. yes g. yes h. no 8-8 b. The measure of an exterior of a equals the sum of the measures of its remote interior s. c. a + b + c = 180º (sum of interior s of a is 180 ), x + c = 180º (straight angle); a + b + c = x + c (substitution) and a + b = x (subtract c from both sides). 8-9 x = 72 and y = = a. congruent (SAS ), x = 79 b. Cannot be determined. c. congruent (AAS ), x 5.9 units d. congruent (SAS ), x a. True b. False (counterexample is a quadrilateral without parallel sides.) c. True d. True e. False (counterexample is a parallelogram that is not a rhombus)

4 WarmUp: In your spiral, show work. January 18, 2017

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6 Quiz Time XXXX Put away phone, close books, HW, etc. You may use only your own INB & pencil. You may use only your own calculator. No sharing. Show work clearly & completely. Turn over quiz when you are finished, & either make a drawing on back of quiz or preview today's lesson. No talking until I've collected all quizzes. To request documented extension, write "more time" & room #, date & time you plan to complete.

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12 "Concave" January 18, 2017

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14 8-5. Answers. a. Triangles (1), (3), and (4) will create convex polygons. They are each isosceles (or equilateral) and the non base angle divides evenly into 360. b. (1): 5, regular pentagon; (3): 18, regular 18 gon; (4): 9, regular nonagon January 18, 2017

15 Objective(s): You Will learn how to find the sum of the interior angles of a polygon; apply this skill to solve problems.

16 Think-Ink-Pair-Share January 18, 2017

17 Think-Ink-Pair-Share January 18, 2017

18 CW: 8-13, 8-14, 8-16 January 18, 2017

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20 INBp94 January 18, 2017

21 a. m = 133 b. x = 120 c. k = 144 d. The two unmarked angles sum to 180 (same side interior angles), so y = 45

22 Add your notes about these quadrilaterals to your Theorem's Toolkit January 18, 2017 INB 95 CPM 481

23 HW: 8-17 to 8-23 (show complete work to support your answers); LL 8-14d (INB 94); MN Special Quad Properties (INB 95/CPM 481); Ch 7 Quiz 1/ answers: a. m = 133 b. x = 120 c. k = 144 d. The two unmarked angles sum to 180 (same side interior angles), so y = 45

24 HW: 8-17 to B = +

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