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4.1 Triangle Sum.notebook November 04, 2014 Agenda for Geometry: • Must complete all missing assessments by the end of the day today. • Pick up a scratch piece of paper and the definition sheet for Chapter 4. • Correct last nights assignment, I will be checking that it is done soon after the bell rings so have it visible. • Triangle Sum Conjecture investigation • First paragraph proof and examples. • Return tools and then work on assignment. • Assignment is 4.1 worksheet not the book problems Nov 68:27 AM Key for Slopes of Parallel and Perpendicular lines Worksheet 19. No, not parallel 20. Yes, parallel 21. Yes, opp reciprocal slopes 22. No, slopes are Nov 68:14 AM For Exercises 1‐4, determine whether each pair of lines through the points given below is parallel, perpendicular, or neither. A(1, 2) B(3, 4) C(5, 2) D(8, 3) E(3, 8) F( −6, 5) 1. AB and BC 2. AB and CD 3. AB and DE 4. CD perpendicular perpendicular neither parallel and EF For Exercises 7‐9, find the slope of each side, and then determine whether each figure is a trapezoid, a parallelogram, a rectangle, or just an ordinary quadrilateral. Explain how you know. 5. Given A(0, − 3), B(5, 3), and Q( −3, −1), find two possible locations for a point P such that PQ is parallel to AB. mAB =6/5 (7,11) Ordinary Quadrilateral (2,5) (8,7) Ordinary Quadrilateral 6. Given C( −2, −1), D(5, −4), and (4, 2), find two possible locations for a point P such that PQ is perpendicular to CD . mCD= 3/7 (7,9) Trapezoid: (1,5) (2,12) Nov 98:47 AM Nov 98:45 AM 10. Quadrilateral HAND has vertices H(−5, −1), A(7, 1), N(6, 7), and D(−6,5). a. Is quadrilateral HAND a parallelogram? A rectangle? Neither? Explain how you know. b. Find the midpoint of each diagonal. What can you conjecture? a. mHA =mND =1/6; m HD = mNA = 6. So, Quad HAND is a rectangle because opposite sides are parallel and adjacent sides are perpendicular. b. Midpoint HN = Minpoint AD = (1/2, 3). The diagonals of the rectangle bisect each other. not 13. Given A( −3, 2), B(1, 5), and C(7, −3), find point D such that quadri ABCD is a rectangle. (3,-6) not Nov 98:47 AM Nov 98:48 AM 1 4.1 Triangle Sum.notebook November 04, 2014 Lesson 4.1 Triangle Sum Conjecture Nov 410:17 AM Nov 31:20 PM We will do and investigation in Pairs. You will need a piece of paper, ruler, and protractor for this investigation. 1) Start by having each group member draw 2 large different triangles on their paper using a ruler. Make sure you have one acute and one obtuse triangle. 2) Next, pass your triangles to another person in your group. 3) Now select a triangle and use a protractor to find the We will need to share scissors, one per every 23 students. measurements of each angle in that triangle. 4) Add these together and share this sum of the three Once all students are seated with above materials in front of them we will get started. Nov 312:56 PM 5) angles with your group. Nov 312:58 PM Now use the other triangle from the sheet and label each corner(vertice) with s, t, and u. s u Triangle Sum Conjecture: t The sum of the measures of the angles in every triangle is ________? 6) Cut out this triangle and wait. 7) Now tear each corner off so you have three good size corners. s u t 8) Line up the corners so the vertices meet at one point. 9) What is the measure of the angle that is created by these three angles Nov 31:21 PM Nov 31:19 PM 2 4.1 Triangle Sum.notebook November 04, 2014 Let's write a paragraph proof to show that the Triangle Sum Conjecture is always true for any triangle. 0 We want to show m<2 + m<4 +m<5 = 180 E C We will first sketch a triangle and use an auxiliary line and answer the questions: • What are we trying to prove? • Why might we draw an auxiliary line to be parallel to one of the sides? • What is the relationship among 1 2 3 If EC is parallel to side AB we know that and 2 4 A 5 5 4 B angles • What congruences can be determined from this diagram? Nov 66:01 AM Sep 284:31 PM Paragraph Proof for Triangle Sum Conjecture: Consider and together as a single angle that forms a linear pair with . E E C C 1 2 3 1 2 3 and Since we will use the substitution property to attain Thus, by the linear pairs conjecture, A 4 5 B A 4 5 B m<2 + m<4 +m<5 = 1800 , as desired. Nov 33:05 PM Sep 284:31 PM Find the measure of each angle indicated EX 1: EX 2: ? ? Nov 53:40 PM Nov 66:24 AM 3 4.1 Triangle Sum.notebook November 04, 2014 EX 4: EX 3: ? ? Nov 66:29 AM EX 5: Nov 66:38 AM Solve for x. EX 6: Nov 66:40 AM Solve for x. Nov 66:44 AM Let's do this Mini-investigation together 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0° 0 1 2 Nov 22:41 PM 3 4 5 Nov 33:34 PM 4 4.1 Triangle Sum.notebook November 04, 2014 Write this onto your definition sheet. Third Angle Conjecture If two angles of one triangle are equal in measure to two angles of another triangle, then the third angles of the triangles are equal in measure. Nov 23:23 PM 5