Section 6: Triangles Part 1


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1 Section 6: Triangles Part 1 Topic 1: Introduction to Triangles Part Topic 2: Introduction to Triangles Part Topic 3: rea and Perimeter in the Coordinate Plane Part Topic 4: rea and Perimeter in the Coordinate Plane Part Topic 5: Triangle Congruence SSS and SS Part Topic 6: Triangle Congruence SSS and SS Part Topic 7: Triangle Congruence S and S Part Topic 8: Triangle Congruence S and S Part Topic 9: Base ngle of Isosceles Triangles Topic 10: Using the Definition of Triangle Congruence in Terms of Rigid Motions Topic 11: Using Triangle Congruency to Find Missing Variables Visit MathNation.com or search "Math Nation" in your phone or tablet's app store to watch the videos that go along with this workbook! 123
2 The following Mathematics Florida Standards will be covered in this section: GCO Explain how the criteria for triangle congruence (S, SS, SSS, and HypotenuseLeg) follow from the definition of congruence in terms of rigid motions. GCO Prove theorems about triangles; use theorems about triangles to solve problems. GGPE Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. GGPE Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. GSRT Prove theorems about triangles. GSRT Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 124
3 Section 6: Triangles Part 1 Section 6 Topic 1 Introduction to Triangles Part 1 We can classify triangles by their angles and their sides. Complete each section of the following table with the most appropriate answers. Can a triangle be both acute and isosceles? Justify your reasoning. Can a triangle be both equiangular and obtuse? Justify your reasoning. Description Representation Name One right angle Three acute angles Let s Practice! 1. Consider the diagram below of an equilateral triangle. One obtuse angle (8xx 13) ft (4xx + 1) ft ll 60 angles Two congruent sides No congruent sides (6xx 6) ft How long is each side of the triangle? Justify your answer. Three congruent sides 125
4 Try It! 2. Consider the triangle below. Let s Practice! 4. Consider the figure below. E F C B D a. If ΔDDDDDD is an isosceles triangle with base DDDD, what is the value of xx? Justify your answer. b. What is the length of each leg? a. fter connecting the points on the plane, Marcos claims that angle BB is a right angle. Is Marcos correct? Explain your reasoning. c. What is the length of the base? 3. How can you determine if a triangle on the coordinate plane is a right triangle? b. How can you classify a triangle on the coordinate plane by its sides? 126
5 Try It! 5. Consider the figure below. C Section 6 Topic 2 Introduction to Triangles Part 2 What is the sum of the measures of the interior angles of a triangle? T C B Formulate how you can prove the sum of measures, if possible. Connect the points on the plane and classify the resulting triangle. Use two different approaches to justify your answer.!"#$%&'!$! )*+,./,0+%&%!20*304+ Triangle Sum Theorem The sum of the interior angles in a triangle is
6 Consider the following figure and complete the following proof. B P Let s Practice! 1. Joan knows the measures of two of the interior angles in a triangle. How could she find the third measure? Explain your reasoning. 2 3 C Try It! Given: and BBBB is parallel to. Prove: mm 1 + mm 2 + mm 3 = 180 Statements 1. is a triangle. 1. Reasons 2. Consider the figure below. K (15xx) 2. BBBB mm 1 + mm 5 = mm PPPPPP mm PPPPPP + mm 4 = mm 1 + mm 5 + mm 4 = J (8xx 11) (3xx + 9) Timothy was trying to find the measure of KK in the triangle above. His answer was 7. He is confused as he cannot understand why mm KK = 7. Is Timothy s answer correct? Justify your answer. L ; mm 2 = mm 4; mm 3 = mm mm 1 + mm 2 + mm 3 =
7 BET THE TEST! 1. Triangle DDDDDD has vertices at DD(5, 8), OO( 3, 10), and GG( 3, 6). Part : Determine what type of triangle DDDDDD is and mark the most appropriate answer. Scalene B Isosceles C Equilateral D Right Part B: If you move vertex DD four units to the left, will the classification of triangle DDDDDD change? If so, what type of triangle will it be? Justify your answer. 2. Stephen is fencing in his triangular garden as shown by the diagram below. X 50 Part : Write an expression for the measure of angle YY. aa Z Y Part B: Stephen measured angle ZZ as 90. He measures angle YY as 38. Did he measure correctly? Justify your answer. 129
8 Section 6 Topic 3 rea and Perimeter in the Coordinate Plane Part 1 Consider the parallelogram below. Consider the rectangle below. h Explain the differences between the perimeter and the area of the rectangle. Each of the smaller squares has a side that is one inch long. What is the perimeter of the rectangle above? What is the formula for finding the area of any parallelogram? Trace the parallelogram above on a separate piece of paper. Try cutting the parallelogram into two triangular pieces. What is the area of the rectangle above? Use your observations to write the formula to find the area of any triangle. What is the formula for finding the perimeter of any rectangle? What is the formula of the perimeter of a triangle? What is the formula for finding the area of any rectangle? 130
9 Let s Practice! 1. Find the area of the following figure. 8 Try It! 2. triangular poster is twice as long as its height. rectangular banner is 3 inches longer than its width. Both the poster and the banner have areas of 648 square inches. a. What is the height and the base of the poster? Justify your answer. 10 b. What is the length and the width of the banner? Justify your answer. 131
10 Section 6 Topic 4 rea and Perimeter in the Coordinate Plane Part 2 How can we find area and perimeter when a figure is on the coordinate plane? Try It! 2. Consider the figure below. yy Let s Practice! D 1. Consider the triangle BBBBBB below. yy xx B G O M T xx Deena s mother is helping her sew a large flag for color guard. Each square on Deena s plan above represents a square foot. a. Determine the amount of fabric Deena need in square feet. a. Which side should be considered the base? Justify your answer. b. Find the area and perimeter of the triangle. b. The flag will be sewn along the edges of the flag. How much ribbon will be needed to the nearest tenth of a foot? 132
11 BET THE TEST! 1. Consider the right triangle below. 2. Dallas is putting down hardwood floors in his home. His living room is pentagonal. Each unit on the coordinate plane represents 5 feet. Find the area of flooring needed in square feet. yy 2xx + 7 5xx 3 E H O xx xx + 4 S U If the perimeter is 40 units, find the value of xx and the area of the triangle. The value of xx is. The area is square units. Which of the following is the total area of the living room? 18 ft^ B 24 ft^ C 450 ft^ D 600 ft^ 133
12 Section 6 Topic 5 Triangle Congruence SSS and SS Part 1 What information do we need in order to determine whether two different triangles are congruent? Let s Practice! 1. If JJJJJJ CCCCCC, finish the following congruence statements and mark the corresponding congruent sides and the corresponding congruent angles. JJJJ JJ CCCC TT KKKK KK When we state triangle congruency, the order of the letters in the names of the triangles is extremely important. J L O C D K C T B E F 2. Complete the congruence statements for the triangles below. I N How can this congruency be stated? R T G TTTTTT II 134
13 Try It! 3. Let s consider the same triangles where TTTTTT GGGGGG. T!"#$%&'!$! )*+,./,0+%&%!20*304+ SideSideSide (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. I G R a. Mark the corresponding congruent sides with hash marks and the corresponding congruent angles with arcs. N We can prove the following triangles are congruent by the SSS Congruence Postulate. Y U X Z T V b. To state that two triangles are congruent, we don t need to know that all three sides and all three angles are congruent. Four postulates help us determine triangle congruency Write the congruency statement for the triangles above. Determine if nglenglengle congruence exists and explain why it does or does not
14 !"#$%&'!$! )*+,./,0+%&%!20*304+ SidengleSide (SS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. We can prove the following triangles are congruent by the SS Congruence Postulate. U Let s Practice! Section 6 Topic 6 Triangle Congruence SSS and SS Part 2 1. What information is needed to prove the triangles below are congruent using the SSS Congruence Postulate? R G C L C T R L Write the congruency statement for the triangles above. 2. What information is needed to prove the triangles below are congruent using the SS Congruence Postulate? M Determine if SideSidengle congruence exists and explain why it does or does not. C O S K D R 136
15 3. What information is needed to prove the triangles below are congruent using the SSS Congruence Postulate? C Try It! 4. Consider CCCCCC and CCCCCC in the figure below. C E S E R Given: CCCC and EEEE Prove: CCCCCC CCCCCC Based on the above figure and the information below, complete the following twocolumn proof. Statements Reasons 1. CCCC 1. Given 2. EEEE 2. Given Reflexive Property of Congruence 4. CCCCCC CCCCCC
16 5. Consider GGGGGG and in the diagram below. BET THE TEST! GG R E 1. Moshi is making a quilt using the pattern below and wants to be sure her triangles are congruent before cutting the fabric. She measures and finds that NNNN VVVV and NN OO. T J Given: RR is the midpoint of and EEEE. Prove: GGGGGG N V O Complete the following twocolumn proof. Statements 1. RR is the midpoint of and EEEE 1. Given Reasons Can Moshi determine if the triangles are congruent with the given information? If not, what other information would allow her to do so? Justify your answer Definition of Midpoint Definition of Midpoint 4. GGGGGG GGGGGG
17 2. Iskra is a structural engineer, designing a tribearing truss for the roof of a new building. She must determine if the triangles below are congruent for the stability of the roof. F Section 6 Topic 7 Triangle Congruence S and S Part 1 Consider the figures below. M I D I Given: FFFF FFFF; FFFF bisects MMMMMM Prove: FFFFFF FFFFFF Which of the reasons for statement 5 is correct? Statements 1. FFFF FFFF 1. Given R Reasons E In the above diagram, WWWWWW FFFFFF based on the S Congruence Postulate. W Name the congruent sides and angles in these two triangles. RR F 2. FFFF bisects MMMMMM 2. Given 3. MMMMMM IIIIII 3. Definition of angle bisector 4. FFFF FFFF 4. Reflective Property of Congruence 5. FFFFFF FFFFFF 5. B SS C SSS D Can t prove congruency!"#$%&'!$! )*+,./,0+%&%!20*304+ nglesidengle (S) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. 139
18 Consider the figures below. M S Consider the triangles below. B W M P N O In the above diagram, MMMMMM SSSSSS based on the S Congruence Postulate. Name the congruent sides and angles in these two triangles. T Identify the postulate you could use to prove that the two triangles are congruent, given each additional congruence statement below. O Congruency Statement Postulate BBBB PPPP OOOO OOOO!"#$%&'!$! )*+,./,0+%&%!20*304+ nglengleside (S) Congruence Postulate If two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of a second triangle, then the two triangles are congruent. 140
19 Consider the figure below. S T Let s Practice! 1. Consider PPPPPP and TTTTTT in the diagram below. P N I I Nadia would like to use the S Congruence Postulate to prove that TTTTTT IIIIII. Would knowing that SS HH be enough information for Nadia to use this postulate? If not, find the missing congruence statement. H T Given: NN and are right angles; II is the midpoint of PPPP Prove: PPPPPP TTTTTT Complete the following twocolumn proof. Statements Reasons 1. NN and are right angles 1. Given 2. NN II is the midpoint of PPPP 3. Given Definition of midpoint 5. TTTTTT PPPPPP TTTTTT
20 Try It! 2. Consider OOOOOO and EEEEEE in the diagram below. Given: OOOO EEEE ; EEEE OOOO Prove: OOOOOO EEEEEE Complete the following twocolumn proof. Statements 1. OOOO EEEE 1. Given Reasons lternate interior angles theorem 3. EEEE OOOO 3. Given 4. PPPPPP HHHHHH Reflexive property 6. OOOOOO EEEEEE 6. H E O P Section 6 Topic 8 Triangle Congruence S and S Part 2 Let s Practice! 1. Consider the figure to the right. Given: PPPP PPPP, LLLL MMMM, MMMM OOOO LLLL bisects MMMMMM, OOOO bisects LLLLLL. Prove: MMMMMM NNNNNN Complete the following twocolumn proof. Statements 1. PPPP PPPP 1. Given 2. LLLL MMMM, MMMM OOOO 2. Given 3. LLLL bisects MMMMMM and OOOO bisects LLLLLL. 3. Given Reasons Definition of lines. 5. mm LLLLLL = mm OOOOOO = Vertical ngles 7. MMMMMM NNNNNN 7. L M P OO NN 142
21 Try It! 2. Consider the figures below. BET THE TEST! 1. Consider the diagram below. yy E P R O C B F G xx E Given: PPPP OOOO; PPPP OOOO V Prove: PPPPPP VVVVVV Select the most appropriate reason for #5. How would you prove GGGGGG by applying ideas of transformations? Statements 1. PPPP OOOO 1. Given 2. PPPP OOOO 2. Given Reasons 3. PPPPPP VVVVVV 3. lternate Interior ngles Theorem 4. EEEEEE OOOOOO 4. Vertical angle theorem 5. PPPPPP VVVVVV 5. S B S C SS D SSS 143
22 2. Consider the figure below. B Section 6 Topic 9 Base ngle of Isosceles Triangles E L By definition, an is a triangle with two equal sides. Consider UUUUUU below. VV U Part : What transformation(s) will prove BBBBBB UUUUUU? Justify your answer. UU Draw the angle bisector VVVV of VV, where PP is the intersection of the bisector and UUUU. Part B: If he knows that EEEE is the angle bisector of BBBBBB, what additional information is needed to prove that BBBBBB UUUUUU using S? Use paragraph proofs to show that mm UU = mm in two ways: by using transformations and triangle congruence postulates. B C D BBBB EEEE BBBBBB EEEEEE EEEEEE EEEEEE EEEEEE UUUUUU Transformations Triangle Congruence Postulates 144
23 Let s Practice! 1. Consider the diagram below. 2. Consider the figure below. R E BB LL KK HH FF GG JJ EE DD D Given: RRRRRR, DDDD is the angle bisector of RRRRRR, and RRRR DDDD C For each of the following congruence statements, name the isosceles triangle and the pair of congruent angles for the triangle based on the diagram above. a. CCCC BBBB CC d. EEEE FFFF Prove: CCCC = CCCC Complete the following twocolumn proof. Statements Reasons 1. RRRR DDDD 1. Given 2. DDDDDD EEEEEE 2. b. FFFF GGGG e. FFFF GGGG Corresponding ngles Theorem 4. DDDDDD DDDDDD 4. c. EEEE Transitive Property Definition of Congruence!"#$%&'!$! )*+,./,0+%&%!20*304+ Base ngle Theorem and its Converse The Base ngle Theorem states that if two sides in a triangle are congruent, then the angles opposite to these sides are also congruent. The converse of this theorem is also true. If two angles of a triangle are congruent, then the sides opposite to these angles are congruent. 7. CCCC = CCCC
24 Try It! 3. Consider the figure below. Q R BET THE TEST! 1. Consider the following TTTTTT, with mm OOOOOO = mm WWWWWW. T M N O P Given: PPPP = PPPP and MMMM QQQQ Prove: PPPP = PPPP Complete the following twocolumn proof. W Waseem was asked to prove that TTTT = WWWW. His work is shown below. There is at least one error in Waseem s work. Describe and explain his error(s). Statements 1. PPPP = PPPP 1. Given 2. PPPPPP is isosceles. 2. Reasons Base ngle Theorem 4. MMMM QQQQ 4. Given 5. NNNNNN RRRRRR and MMMMMM QQQQQQ Transitive Property Definition of Congruence 8. PPPP = PPPP
25 Section 6 Topic 10 Using the Definition of Triangle Congruence in Terms of Rigid Motions How can rigid motion(s) be used to determine congruence? Let s Practice! 1. Consider the diagram below. yy EE Ø Rigid motions move figures to a new location without altering their or, thus maintaining the conditions for the figures to be congruent. By definition, two figures are if and only if there exists one, or more, rigid motions which will map one figure onto the other. SS OO PP xx Consider the diagram below. yy TT JJ BB Find a rigid motion that will map SSSSSS onto OOOOOO. PP xx TT OO Find a rigid motion that will map PPPPPP onto JJJJJJ. Justify the use of the SSS Congruence Postulate to prove that PPPPPP JJJJJJ. 147
26 Try It! 3. Consider the diagram below. 2. Consider the diagram below. II yy EE rr UU TT OO KK YY VV HH xx CC a. Find a rigid motion that will map onto CCCCCC. b. Suppose that mm = , = 6.4, and VVVV = Justify the use of the SS Congruence Postulate to prove that CCCCCC. Which of the following rigid motions will map TTTTTT onto KKKKKK? Select all that apply. o reflection over the yyaxis o rotation of 270 clockwise about the origin o The translation (xx, yy) (xx + 5, yy 1) o reflection over the line yy = xx followed by a rotation of 180 clockwise about the origin o The translation (xx, yy) (xx 5, yy + 1) followed by a rotation of 90 counterclockwise about vertex YY c. Suppose that your friend suggests a translation as the rigid motion that maps onto CCCCCC. Is your friend correct? Justify your answer. 148
27 BET THE TEST! 1. Consider the diagram below. yy Section 6 Topic 11 Using Triangle Congruency to Find Missing Variables Consider the figures below xx 8 PP CC xx Find the value of xx in order to prove that the two triangles are congruent by the SS Congruence Postulate. Justify your work. TT OO PP Let s Practice! 1. Consider the figures below. Find a combination of rigid motions that will map CCCCCC onto PPPPPP and determine if CCCCCC PPPPPP. 2yy 3 yy + 5 (4yy + 11) Find the value of yy in order to prove that the two triangles are congruent using the S Congruence Postulate. Justify your work. 149
28 Try It! BET THE TEST! 2. Consider the figure below. xx 3 2xx 7 1. Consider the figure below. SS 5xx + 6 yy 4 12 PP 4yy 4xx 50 RR 20 TT 21 Find the values of xx and yy that prove the two triangles are congruent using the SSS Congruence Postulate. QQ 3. Consider the figure below. Part : If PPPP TTTT and PPPP TTTT, which triangle congruency postulate can we use to determine PPPPPP TTTTTT given the information on the figure? Select all that apply xx 9 (7yy 4) o o o o o S S SS SSS SS 17 Part B: What are the values of xx and yy? Find the values of xx and yy that prove the two triangles are congruent using the S Congruence Theorem. Justify your work. Test Yourself! Practice Tool Great job! You have reached the end of this section. Now it s time to try the Test Yourself! Practice Tool, where you can practice all the skills and concepts you learned in this section. Log in to Math Nation and try out the Test Yourself! Practice Tool so you can see how well you know these topics! 150
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