Day 31 Bellringer. (a) 2 and 7. (b) 3 and 7. (c) 3 and 6. Page 1

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Jemimah Wilkinson
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1 Day 31 Bellringer 1. In the figure below lines PQ and RS are parallel. State whether the following angles are corresponding, alternate interior or alternate exterior angles. P R Q S (a) 2 and 7 (b) 3 and 7 (c) 3 and 6 HighSchoolMathTeachers@2018 Page 1

2 Day 31 Bellringer 2. Find the alternate interior and alternate exterior angles represented by letters. The two lines shown in each case are parallel. (a) 113 x y (b) 31 x y HighSchoolMathTeachers@2018 Page 2

3 Day 31 Bellringer Answer Keys Day 31: 1. (a) Alternate interior angles (b) Corresponding Angles (c) Alternate exterior angles 2. (a) x = 113, y = 67 (b) x = 31, y = 149 HighSchoolMathTeachers@2018 Page 3

4 Day 31 Activity 1. On the plain paper provided, draw two straight lines of about 5 inches each on either side of the ruler to form a pair of parallel lines as shown below. Label the two lines as PQ and RS as shown below. P Q R 2. Mark a point, A, on line PQ at the position shown below. P A Q S R S 3. Similarly, mark two points, B, and C on line RS in the positions shown below. P A Q R B 4. Now, using a ruler draw two lines, from point A to point C and from point A to point B respectively as shown below. C S P A Q R B C S HighSchoolMathTeachers@2018 Page 4

5 Day 31 Activity 5. What name is given to the plane figure formed between points A, B, and C? 6. Consider the following angles: BAC, ABC, and ACB. Are these angles located within or outside the plane figure you have identified in 5 above? 7. Measure these three angles accurately using a protractor: BAC, ABC, and ACB and write down their measures. 8. What is the sum of these three angles? Do you think this is usually the case for any plane figure of the same kind? HighSchoolMathTeachers@2018 Page 5

6 Day 31 Activity In this activity, students will work in groups of four to discover that when the interior angles of a triangle are summed up, they add up to 180. Students in the respective groups will require a pencil, a plain paper, a ruler and a protractor. It is assumed from the foregoing that students are comfortable with measuring angles using a protractor. Answer Keys Day 31: 1. No response 2. No response 3. No response 4. No response 5. Triangle 6. Located within the plane figure 7. Each of the three angles will vary from group to group depending on the triangle they have come up with. 8. The sum should be accurately 180. This is a question to enable the students think of any other triangle having interior angles summing up to 180 though most of them will agree. HighSchoolMathTeachers@2018 Page 6

7 Day 31 Practice Use the figure below to answer question 1-5. RS PQ and the two transversal lines intersect the pair of parallel lines to form triangle ABC as shown below. R 59 A S x y z P 44 B C Q 1. Find the measure of y 2. Find the measure of z 3. Find the measure of x 4. Find the sum of x, y and z. 5. What is your conclusion about the sum in question 4 in relation to the triangle formed above? HighSchoolMathTeachers@2018 Page 7

8 Day 31 Practice Use the figure below to answer questions Triangle PQR is formed between the parallel lines. A 29 P B x y z C 81 Q R D 6. Find the measure of y 7. Find the measure of CQP 8. Find the measure of BPQ 9. Find the measure of BPR 10. Find the measure of z 11. Find the measure of x HighSchoolMathTeachers@2018 Page 8

9 Day 31 Practice 12. Find the sum of the interior angles x, y and z in the figure above. 13. What do you discover about the sum in question 12 with respect to the triangle formed above? Use the figure below to answer questions FG HI and the two transversal lines intersect the pair of parallel lines to form triangle XYZ. F X 28 G a 111 b c H Y Z I 14. Find the measure of FXY 15. Find the measure of FXZ 16. Find the measure of b 17. Find the measure of a HighSchoolMathTeachers@2018 Page 9

10 Day 31 Practice 18. Find the measure of c 19. Find the sum of these angles: a, b and c 20. What does this tell you about the three angles in relation to triangle XYZ? HighSchoolMathTeachers@2018 Page 10

11 Day 31 Practice Answer keys Day 31: 1. y = z = x = The sum of interior angles of the triangle add up to y = z = x = The sum of interior angles of the triangle add up to b = a = c = The sum of interior angles of the triangle add up to 180 HighSchoolMathTeachers@2018 Page 11

12 Day 31 Exit Slip In the figure below, AB CD and the two transversal lines intersect the pair of parallel lines to form triangle JKL as shown below. 62 A J 47 B a c b C K L D (a) Find the measure of a (b) Find the measure of b (c) Find the measure of c (d) Find the sum of a, b and c. What is your conclusion about that sum in relation to the triangle formed above? HighSchoolMathTeachers@2018 Page 12

13 Day 31 Exit Slip Answer Keys Day 31: (a) a = 62 (b) b = 47 (c) c = 71 (d) a + b + c = = 180. The interior angles of the triangle add up 180 HighSchoolMathTeachers@2018 Page 13

14 Day 32 Bellringer In the figure below, triangle PQR is formed between the pair of parallel lines AB and CD. A 22 P B x y z C 79 Q R D (a) Find the measure of y (b) Find the measure of BPR (c) Find the measure of x (d) Find the measure of z (e) Compare the measure of x to that of y, is there any relationship between the two angles? HighSchoolMathTeachers@2018 Page 14

15 Day 32 Bellringer Answer Key Day 32: (a) y = 79 (b) BPR = 22 (c) x = 79 (d) z = 22 (e) x = y = 79 ; They are congruent. HighSchoolMathTeachers@2018 Page 15

16 Day 32 Activity 1. Label the plain paper provided as ABCD. 2. Fold the paper carefully at its center making sure the edge AB is aligned exactly on the edge CD as shown below. A B A B C C D D 3. Using the pair of scissors provided, cut the folded paper carefully along the diagonal as shown below. 4. Remove the outer cut-out and open up the paper as shown below. What type of plane figure is formed from the other cut-out? P B D HighSchoolMathTeachers@2018 Page 16

17 Day 32 Activity 5. Label the remaining vertex as P. 6. Using a ruler, measure BP and DP in inches on the cut out. What do you notice? 7. Using a protractor, measure angles BDP and DBP and compare their measures. What do you notice? HighSchoolMathTeachers@2018 Page 17

18 Day 32 Activity In this activity, the students will work in groups of four to discover that an isosceles triangle has at least two equal sides and consequently, its base angles are congruent through simple paper folding and cutting. The students in the respective groups will require an A5 size plain paper, a protractor, a pair of scissors and a ruler. Emphasize that each procedure to be done carefully to guarantee accurate observations. Answer Keys Day 32: 1. No response 2. This is to ensure that the fold is straight and it connects the midpoint of edge AC to the midpoint of edge BD. 3. No response 4. Triangle 5. No response 6. BP = DP 7. BDP = DBP HighSchoolMathTeachers@2018 Page 18

19 Day 32 Practice Use the figure below to answer questions 1-5. Isosceles triangle ABC is formed by the parallel lines XY and PQ. CAY = ABC = θ. 1. Find the measure of BAX in terms of θ. 2. Find the measure of ACB in terms of θ. 3. Find the measure of BAC in terms of θ. 4. Express the sum of angles ABC, ACB and BAC. 5. Compare the measures of ABC to ACB. What do you notice? HighSchoolMathTeachers@2018 Page 19

20 Day 32 Practice Use the figure below to answer questions The parallel lines JK and LM together with the two transversal lines form triangle PQR as shown below. JPQ = KPR = α. 6. Find the measure of QPR in terms of α. 7. Find the measure of PQR in terms of α. 8. Find the measure of PRQ in terms of α. 9. Identify two congruent interior angles in triangle PQR. 10. Hence, identify two equal edges on triangle PQR. 11. Using your responses from questions 9 and 10, give the type of triangle PQR. HighSchoolMathTeachers@2018 Page 20

21 Day 32 Practice Use the figure below to answer questions Triangle XYZ is formed between the parallel lines FG and JK. MXF = GXN = β. 12. Find the measure of MXN in terms of β. 13. Hence, find the measure of YXZ in terms of β. 13. Find the measure of FXY in terms of β. 14. Find the measure of GXZ in terms of β. 15. Using the measure of MXF, find the measure of XZY. 16. Using the measure of GXN, find the measure of XYZ. 17. Compare XZY to XYZ. What is the relationship between the two angles? 18. Using the relationship between angles XZY and XYZ, identify two equal line segments from triangle XYZ above. 19. Using information from questions 17 and 18 above, what type of triangle is triangle XYZ? 20. Give the major reason to support your response to question 19 above. HighSchoolMathTeachers@2018 Page 21

22 Day 32 Practice Answer keys Day 32: 1. θ 2. θ θ ABC = ACB = θ; The two angles are congruent α 7. α 8. α 9. PQR and PRQ 10. PQ and PR 11. Isosceles β β 14. β 15. β 16. β 17. XZY = XYZ; The two angles are congruent 18. XY and XZ 19. Isosceles 20 The base angles are equal; XZY = XYZ HighSchoolMathTeachers@2018 Page 22

23 Day 32 Exit Slip In the figure below isosceles triangle, JKL is formed between the parallel lines AB and CD. AJK = AJM = β. M A β J B β C K L D (a) Find the measure of JKL in terms of β. (b) Find the measure of JLK in terms of β. (c) Compare the measures of angles JKL and JLK. What do you discover? HighSchoolMathTeachers@2018 Page 23

24 Day 32 Exit Slip Answer Keys Day 32: (a) JKL = β (b) JLK = β (c) JKL = JLK = β ; The two angles are congruent. HighSchoolMathTeachers@2018 Page 24

25 Day 33 Bellringer 1. ABC is an isosceles triangle. Line AB = 3 inches and line AC = 5 inches. Use the triangle to answer the questions that follow. C a) Find the length of AD. A D B b) What is the length of BD? c) Find the length of the side BC d) If CAD = 31, What is the size of CBD 2. A line DF is 9 inches long. If G is its midpoint, find the size of line GF. HighSchoolMathTeachers@2018 Page 25

26 Day 33 Bellringer Answer Key Day 4 1. a) 1 5 inches b) 1 5 inches c) 5 inches d) inches HighSchoolMathTeachers@2018 Page 26

27 Day 33 Activity 1. Draw a line of length 4 inches and name it AB. 2. Place the protractor on line AB as shown the measure angle60. A B 4. Remove the protractor and join point A and the mark with a straight line as shown below. A B 5. Measure a distance of 4 inches along the line you have drawn in 4 above, mark it and label it C as shown. 6. Using a pencil and a ruler, join C and B with a straight line to get ABC below HighSchoolMathTeachers@2018 Page 27

28 Day 33 Activity 7. Mark the midpoints of sides AB, AC and BC as D, E and F respectively. C E F A B D 8. Join the A and F, C and D then B and E as shown below. C E F A B D What do you notice about the intersection of the midpoints? HighSchoolMathTeachers@2018 Page 28

29 Day 33 Activity In this activity, students are required to draw an equilateral triangle and its medians to study where and how they meet. Students are required to work in groups of at least three. Each group is required to have a plain paper, a pencil, a protractor and a ruler. Answer Keys Day 33: 1-7. No response 8. They all meet at the same point HighSchoolMathTeachers@2018 Page 29

30 Day 33 Practice Use the information below to answer questions 1-5. In STR, ST = 11 inches, TR = 8 inches and RS = 10 inches. The line segments AT, CR and BS are the medians of STR. C T S B A R 1. What is the length of BR? 2. Find the length of SC 3. What is the length of AR? 4. What is the length of AS? 5. Find the length of BT. In questions 6 to 10. State whether the given statement is true or false. 6. A midpoint of a line divides it into two equal parts. 7. Medians of a triangle meet at two different points. 8. Medians of a triangle meet at one of its vertices. 9. Median of a triangle must run from one vertex and meet the opposite side at a right angle. HighSchoolMathTeachers@2018 Page 30

31 Day 33 Practice 10. Medians of a triangle meet at one point. 11. Draw the medians of the triangle given below. Use the triangle below to answer questions The Line segments AN, MB amd CO are the medians of MNO. F A B O M C N 12. Write an equation that relates AM and AF. 13. Write an equation that can relate AM and FM. 14. If FB is 3 5 inches long. What is the length of the side FN. HighSchoolMathTeachers@2018 Page 31

32 Day 33 Practice 15. Write an equation that relates MC and CN. 16.Write an equation that relates MC and MN. 17. Which name is given to the point marked F? Use the triangle below to state whether the statements given questions is true or false. CF, BD and AE are the medians of the triangle. C D E O A 18. AB = 1 2 AF F B 19. CDO must be equal to CE EB HighSchoolMathTeachers@2018 Page 32

33 Day 33 Practice Answer Key Day inches inches 3. 5 inches 4. 5 inches 5. 4 inches 6. True 7. False 8. False 9. False 10. True AM = AF 13. FM = 2AM Inches 15. MC = CN 16. MN = 2MN 17. Centroid 18. False 19. False 20. False HighSchoolMathTeachers@2018 Page 33

34 Day 33 Exit Slip 1. Draw the medians of the following triangle identifying any resultant features created as a result of drawing the medians. R S T HighSchoolMathTeachers@2018 Page 34

35 Day 33 Exit Slip Answer Keys Day 33: R S T HighSchoolMathTeachers@2018 Page 35

36 Day 34 Bellringer 1. If a triangle is symmetrical about = x, is their any rigit motion that can be realized in any form of setup created by the triangle. Which one, explain. 2. One of the base angles of an Isosceles triangle is 40, find the size of all other angles. 3. Can we have a base angle of an Isosceles triangle being 90? Explain your answer. 4. Explain why one can view an equilateral triangle as an Isosceles triangle. 5. Two triangles with a common side have two corresponding vertices at equal distance from the common side. What is the condition that such kind of set up will make a bigger triangle when the common side is deleted? HighSchoolMathTeachers@2018 Page 36

37 Day 34 Bellringer Answer Key Day Yes, reflection. One side of the triangle below y = x is a reflection of the one above it or vise versa and No because the sum of the two base angles would be 180 before the third one is added. This cannot be since the sum of all interior angles must be An equilateral triangle has at least two equal angles and sides. 5. The two corresponding vertices must be on the same line as one of the ends of the common side. HighSchoolMathTeachers@2018 Page 37

38 Day 34 Activity 1. On a plain paper, draw a straight line of length between 2 and 4 inches. 2. From one side of the drawn line, draw another line of the same length to meet at an acute angle 4. Connect the ends of the two lines to form a triangle. 5. Confirm that the triangle formed in an Isosceles triangle by measuring and writing their lengths. 6. Label the vertices. 7. Identify the baseline and the base angles. 8. Measure the all interior angles. 9. Sum the angles, what do you get? 10. What is common about the measurements of the base angles? 11. Draw the medians of the lines, what do you notice? HighSchoolMathTeachers@2018 Page 38

39 Day 34 Activity In this activity, students are required to draw an Isosceles triangle then prove that the base angles are equal equal, the sum of interior angles is 180 and the medians intersect at a common point. Students will work in groups of 5. Each group will be provided with a plane paper, a pencil, a ruler and a protractor. Answer Keys Day 34: 1 4. No response 5. Answers will vary but two must sides must be approximately equal 6. No response 7-8. Answers may vary They are equal 11. They intersect at the same point HighSchoolMathTeachers@2018 Page 39

40 Day 34 Practice Use the information below to answer questions 1-7. In the figure below, lines HE and AD are parallel. Angle FGE = 61 and angle FGC = 97. F H G E A B C D 1. Find the size of angle EGC 2. Find the size of angle GCB 3. Find the size of angle HGB 4. Find the size of angle ABG 5. Find the size of angle GBC 6. Find the size of angle BGC 7. Find the sum of angles GBC, BCG and CGB. HighSchoolMathTeachers@2018 Page 40

41 Day 34 Practice Use the information below to answer questions Consider the following triangle. 8. Identify the type of triangle. 9. Identify the points in each line that divides it into two equal parts. 10. Draw all the medians of the triangle. 11. Identify the number of points where they meet. 12.Identify if the point(s) above are inside or outside the triangle. Page 41

42 Day 34 Practice Use the information below to answer questions In the figure below, line ML = KL and MN = NK. KM = 24 in and ML = 20 in and angle MLN = 37. M N L K 13. Find the size of line NK. 14. Find the size of line KL. 15. Identify any two pairs of complementary angles. 16. Find the size of angle KLN. 17. Find the size of angle NKL 18. Find the size of angle NML 19. Compare the size of angles in 17 and 18 above. 20. Identify, if any, a rigid motion between triangles MNL and NKL. HighSchoolMathTeachers@2018 Page 42

43 Day 34 Practice Answer Key Day Scalene Triangle 9. Answers may vary One 12. Inside in in 15. Angles NML and MLN Angles NLK and LKN The angles are equal to Reflection Page 43

44 Day 34 Exit Slip A triangle is symmetrical about x axis. Identify the y codinate of the point at which the medians meet. Explain why. HighSchoolMathTeachers@2018 Page 44

45 Day 34 Exit Slip Answer Keys Day 34: The y coordinate is 0. This is because of one of the medians willlie of the x axis whose equation is y = 0. Since thepoint of intersection is shared among all the medians, it must satisfy the equation of each median including y = 0, thus, the y coordinate. HighSchoolMathTeachers@2018 Page 45

6 CHAPTER. Triangles. A plane figure bounded by three line segments is called a triangle.

6 CHAPTER. Triangles. A plane figure bounded by three line segments is called a triangle. 6 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius A plane figure bounded by three line segments is called a triangle We denote a triangle by the symbol In fig ABC has