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Foundations of Math 2 Unit 3 : Triangles Notes & Homework Mrs. Bello Fall 2016 1 Foundations Math II Name_____________________ Pd ____ Unit 3: Triangles Day 1 Lesson Homework Stamp Triangle Sum 2 Isosceles Triangle 3 Isosceles Triangle 4 Dilations & Similarity 5 Quiz #1 6 Mid-segments 7 Triangle Congruence 8 Triangle Congruence 9 Quiz #2 10 Flow Charts Proofs 11 Flow Charts Proofs 12 Review 13 Test #3 – Triangles 2 Foundations of Math 2 – Unit 3: Triangles Day 1: Triangle Sum Theorem Warm-Up Grab a colored sheet of paper. I want you to use a ruler and draw a triangle. Label the interior (inside) angles (∠1, ∠2, ∠3). Now cut your triangle out. Now we are going to tear the 3 corners of our triangle off. I want you to line them up on the following line. What do you notice? Triangle Sum Theorem: _________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ 3 You try! Solve for the missing angle. 1. 2. Solve for x. 3. 4. Linear Pairs: ________________________________________________________________________________ ___________________________________________________________________________________________ x 76o You try! Find x, then y. y x 126 o Vertical Angles: _____________________________________________________________________________ ___________________________________________________________________________________________ 4 Practice: Name____________________ 5 6 Day 2 – Isosceles Triangles Parts of a Triangle: Triangle – a three-sided polygon Name – Sides – Vertices – Angles – Classifying Triangles by Angles: Acute ∆ Obtuse ∆ Right ∆ Isosceles ∆ Equilateral ∆ Equiangular ∆ - Classifying Triangles by Sides: Scalene ∆ Example #1: Find x and the measure of each side of equilateral triangle RST. 7 Example #2: Find x, JM, MN, and JN if ∆JMN is an isosceles triangle with JM MN . Isosceles Triangle: A triangle with at least __________ sides congruent. Isosceles Triangle Theorem: If two sides of a triangle are ____________________, then the angles opposite those sides are ____________________. Ex: Example #3: If DE CD , BC AC , and mCDE 120 , what is the measure of BAC ? 8 Theorem: If two angles of a _______________ are congruent, then the sides opposite those angles are ___________________. Ex: Example #4: a.) Name all of the congruent angles. b.) Name all of the congruent segments. A triangle is _____________________ if and only if it is ___________________. Each angle of an equilateral triangle measures ________. Example #5: ∆EFG is equilateral, and EH bisects E . a.) Find m1 and m2 . b.) Find x. 9 10 Day 3: Isosceles Triangles cont. Warm-Up 1. 2. 32o 70o a 60o x b 100o 3. (x,y)(x+2, -y) c d e f 4. Reflect over y= -x. A A C C B A’_________ B’_________ C’__________ B A’_________ B’_________ C’__________ 11 Find x and y 1) 2) 3) 4 3 4 4 3 4) 5) Equilateral Triangle 4x 40 60 6) Equilateral Triangle 10 y Find x, y and z 7) 8) 5z 9) 9 9 3x 4y 9 12 Find x (and y) in each figure below. 1. 2. 3. x x 4. 130o 5. 6. 13 14 Day 4: Dilation and Similarity Warm-Up 3. Dilate the figure with respect to the origin by a scale factor of ½. List the preimage and image points. 4. Dilate the figure with respect to the origin by a scale factor of 1/4 . List the preimage and image points. 15 Similarity Practice Solve for x. 1. If 10. x 8 x y , then . y 3 8 3 x 9 6 24 2. If 11. x 8 x 8 y 3 , then . y 3 8 3 4 3 y 3 y 3 12. 5 3 2y 7 y 3 Solve for the variable. 13. MN:MO is 3:4 14. PQR side lengths: STU side lengths is 1:3 M x P S 5 x 9 R O N U 36 12 Q T Use the diagram and the given information to find the unknown length. 15. Given AB AE , find BC. BC ED 16. Given AB AE , find BC. BC ED . 16 Notes: Similar Polygons Similar Polygons: SYMBOL for SIMILAR: ________ Corresponding angles are _____________________________________________ Corresponding sides are _______________________________________________ Writing Similarity Statements: Corresponding <’s: Proportional Sides: A BC XYZ If 2 polygons are _____________, then the ratio of the lengths of 2 corresponding sides is called the ___________________. What is the scale factor of ∆ABC to ∆XYZ? ________________ Practice: 1.) If polygon LMNO ~HIJK , completing proportions and congruence statements. Hint: Draw a diagram!! a. M __?__ d. MN IJ ? JK b. K __?__ e. HK HI ? LM c. N __?__ f. IJ HK MN ? 17 2.) In the diagram, polygon ABCD ~ GHIJ. A 11 D 8 x x B G y H 5.5 11 C a. Find the scale factor of polygon ABCD to polygon GHIJ. c. Find the values of x and y. J 8 I b. Find the scale factor of polygon GHIJ to polygon ABCD. d. Find the perimeter of each polygon. e. Find the ratio to the perimeter of ABCD to perimeter of GHIJ. If 2 polygons are ___________, then the ratio of their perimeters is equal to the ratios of their ___________. 3.) The ratio of one side of ∆ABC to the corresponding side of similar ∆DEF is 3:5. The perimeter of ∆DEF is 48in. What is the perimeter of ∆ABC? 18 Classwork (# 1 – 9) Show proportions for all problems. 1. Parallelogram EFGH is similar to parallelogram WXYZ. X F G Y 6 in 2 in E 3 in H W Z What is the length of WZ ? a) 3 in b) 6 in c) 7 in d) 9 in 2. Lance the alien is 5 feet tall. His shadow is 8 feet long. 5 ft 32 ft 8 ft At the same time of day, a tree’s shadow is 32 feet long. What is the height of the tree? a) 20 feet b) 24 feet c) 29 feet d) 51 feet X ? 3. Pentagon JKLMN is similar to pentagon VWXYZ. What is the measurement of angle X? L a) 30 b) 60 c) 150 60 K 150 4. Triangle LMN is similar to triangle XYZ. Y W d) 120 J M N V Z 19 L Z 18 feet 24 feet 8 feet Y N 12 feet X M What is the length of YX ? a) 2 feet b) 3 feet c) 4 feet d) 6 feet 5. Triangle PQR is similar to triangle DEF as shown. E Q 6 cm 4 cm P 6 cm R D 9 cm F Which describes the relationship between the corresponding sides of the two triangles? a) PQ 4 DE 6 b) PQ 6 DE 4 c) PQ 4 EF 9 d) PR 6 DE 6 1 6. A six-foot-tall person is standing next to a flagpole. The person is casting a shadow 1 feet 2 in length, while the flagpole is casting a shadow 5 feet in length. How tall is the flagpole? a) 30 ft b) 25 ft c) 20 ft d) 15 ft 20 7. PQR is similar to XYZ. Y Q 6 5 30 R P Z 10 X What is the perimeter of XYZ? a) 21 cm b) 63 cm c) 105 cm d) 126 cm 8. The shadow cast by a one-foot ruler is 8 inches long. At the same time, the shadow cast by a pine tree is 24 feet long. a) 3 feet b) 16 feet c) 36 feet d) 192 feet 1 foot What is the height, in feet, of the pine tree? 8 inches 9. If triangles ADE and ABC shown in the figure to the right are similar, what is the value of x? 24 feet 12 C B 4 a) 4 b) 5 c) 6 d) 8 e) 10 x E D 2 3 A 21 Day 5 (Quiz Day) : Mid-segments Warm-Up . e 45 44 a=______ b=______ c=______ d=______ e=______ f=______ g=______ h=______ d a 51 b f c g 45 h 22 23 24 25 26 27 Day 6: Midsegments cont. Warm-Up a=_________ 1. 2. 31 x=_______ b=_________ a 109 b c=_________ d=_________ 2x-9 36 46 3x+8 c e=_________ d 19 e 3. e 39 45 o 36 44 o a=______ b=______ c=______ d=______ e=______ f=______ g=______ h=______ d 48 a o 51 b f c 45 51o g h 28 1. x 18 2. 5x 70 3. 84 3x 4. x-1 45 5. 5x-2 4 29 For 6-11, name the segment that is parallel to the given segment. A F C 6. AB 7. EF 8. GE 9. BC 10. CA 11. FG G E B 12. Find the values of x and y. 3x-6 y x 2x+1 30 Day 7: Triangle Congruence 31 Congruent Figures: have the same __________________ & same _______________________. Each _____________________ (“matching”) side and angle of congruent figures will also be _______. Example #1: (Points can be named in any consecutive order, but each corresponding vertex must be in the same order for each figure). ABCDE ≅ VWXYZ Congruent Angles Congruent Sides Example #2: Given: ABCD≅ EFGH. Complete the following a) Rewrite the congruence statement in at least 2 more ways. b) Name all congruent angles c) Name all congruent sides We will deal mostly with congruent triangles. Two triangles are congruent if and only if their vertices can be matched up so that the _______________________________ (both angles & sides) are congruent. Example #3: Given: ∆𝐻𝐼𝐽 ≅ ∆𝑀𝑁𝑂. Name all congruent sides and all congruent angles. Write the congruence statement in at least 2 more ways. 32 Example #4: If ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍, and 𝑚∠𝐴 = 3𝑥 + 12, 𝑚∠𝐵 = 3𝑥 + 1 and 𝑚∠𝑋 = 𝑥 + 44, find x. What if we wanted to prove that 2 polygons were congruent? What would we need to do? Because triangles only have three sides, we can take some shortcuts… I. If all three sides are given, we call this ___________. SSS Postulate: If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent. II. If 2 sides and the angle BETWEEN those sides are given, we call this _________. SAS Postulate: If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent. Included means _______________ III. If 2 angles and the side BETWEEN those angles are given, we call this _________. ASA Postulate: If 2 angles and the included side are congruent to 2 angles and the include side of another triangle, then the triangles are congruent. IV. If 2 angles and the side NOT BETWEEN those angles are given, we call this _______. AAS Postulate: If 2 angles and their non-included side are congruent to 2 angles and the non-included side of another triangle, then the triangles are congruent. 33 Example #5: Are the triangles congruent? If so, why? Anytime that 2 triangles share a side, think _________________ property! ̅̅̅̅, EF ̅. ̅ ≅ GH ̅̅̅̅ ≅ HF ̅̅̅̅ and G is the midpoint of GI Example #6: IE 34 35 Triangle Congruence Picture Questions I. If the triangles can be proven congruent, give the reason (SSS, SAS, ASA, or AAS). If there is not enough information to prove the triangles congruent, write “none.” 1. 5. 9. 6. 10. 7. 11. 8. 12. 2. 3. 4. II. Determine whether you can conclude that another triangle is congruent to ∆ABC. If so, complete the congruence statement and give the reason (SSS, SAS, ASA, or AAS). If not, write “none.” A 1. A C C 2. B B K D 3. P B N Y ∆𝐴𝐵𝐶 ≅ ∆_____________ by ______________ C ∆𝐴𝐵𝐶 ≅ ∆_____________ by ______________ A ∆𝐴𝐵𝐶 ≅ ∆_____________ by ______________ 36 4. A B 5. S X B 6. A C A Z B Y C J C ∆𝐴𝐵𝐶 ≅ ∆_____________ by ______________ 7. ∆𝐴𝐵𝐶 ≅ ∆_____________ by ______________ 8. ∆𝐴𝐵𝐶 ≅ ∆_____________ by ______________ 9. P A C P A 60 B 61 C B C B Q D 60 A Q ∆𝐴𝐵𝐶 ≅ ∆_____________ by ______________ ∆𝐴𝐵𝐶 ≅ ∆_____________ by ______________ ∆𝐴𝐵𝐶 ≅ ∆_____________ by ______________ For Question #1-4, 𝜟𝑭𝑰𝑵 ≅ 𝜟𝑾𝑬𝑩 1. Name the three pairs of corresponding sides. _______________ 2. Name the three pairs of corresponding angles ______________ 3. Is it correct to say 𝑭 ≅ 𝜟𝑩𝑬𝑾 ?_____ Explain why?_________________________________________ 4. Is it correct to say 𝑭 ≅ 𝜟𝑬𝑩𝑾 ?_____ Explain why?_________________________________________ For questions #5-9, use the congruent triangles to the right 5. 𝛥𝐴𝐵𝑂 ≅ 𝛥________ 6. ∡𝐴 ≅ ______ ̅̅̅̅ ≅ ______ 7. 𝐴𝑂 8. 𝐵𝑂 = ______ C D O A B 37 What additional information is required in order to know that the triangles are congruent by the given reason? D 1. ASA 5. SAS R I J U T S 2. SAS X 6. ASA K S L W S T H M V T K U 3. SAS C 7. SSS L R Q B K A S J T 4. ASA 8. SAS D F U J L W K V M E K 38 Day 8: Triangle Congruence cont. Warm-Up Draw and label a diagram. Then solve for the variable and the missing measure or length. 1. If ∆𝐵𝐴𝑇 ≅ ∆𝐷𝑂𝐺, and 𝑚∠𝐵 = 14, 𝑚∠𝐺 = 29, 𝑎𝑛𝑑 𝑚∠𝑂 = 10𝑥 + 7. Find the value of x 𝑚∠𝑂. x = ___________ 𝑚∠𝑂= _________ 2. If ∆𝐶𝑂𝑊 ≅ ∆𝑃𝐼𝐺, and 𝐶𝑂 = 25, 𝐶𝑊 = 18, 𝐼𝐺 = 23, 𝑎𝑛𝑑 𝑃𝐺 = 7𝑥 − 17 . Find the value of x and PG. x = ___________ PG=___________ 3. If ∆𝐷𝐸𝐹 ≅ ∆𝑃𝑄𝑅 and 𝐷𝐸 = 3𝑥 − 10, 𝑄𝑅 = 4𝑥 − 23, 𝑎𝑛𝑑 𝑃𝑄 = 2𝑥 + 7. Find the value of x and EF. x = ___________ EF = __________ 39 I. Use the given information and triangle congruence statement to complete the following. 1. ∆𝐴𝐵𝐶 ≅ ∆𝐺𝐸𝑂, AB = 4, BC = 6, and AC = 8. What is the length of ̅̅̅̅ 𝐺𝑂? How do you know? 2. ∆𝐵𝐴𝐷 ≅ ∆𝐿𝑈𝐾, 𝑚∠𝐷 = 52°, 𝑚∠𝐵 = 48°, 𝑎𝑛𝑑 𝑚∠𝐴 = 80°. a. What is the largest angle of ∆𝐿𝑈𝐾? b. What is the smallest angle of ∆𝐿𝑈𝐾? 3. ∆𝑆𝑈𝑁 ≅ ∆𝐻𝑂𝑇. or why not. ∆𝑆𝑈𝑁 is isosceles. Is there enough information to determine if ∆𝐻𝑂𝑇 is isosceles? Explain why 40 II. 1. Complete the congruence statement for each pair of congruent triangles. Then state the reason you are able to determine the triangles are congruent. If you cannot conclude that triangles are congruent, write “none” in the blanks. ∆𝐸𝐹𝐷 ≅ ∆___________ ∆𝐴𝐵𝐶 ≅ ∆___________ 2. by ________ 3. ∆𝐿𝐾𝑀 ≅ ∆___________ by ________ E by ________ L A D F B H C K M T G J 4. ∆𝐴𝐵𝐶 ≅ ∆___________ ∆𝐴𝐵𝐶 ≅ ∆___________ 5. by ________ A by ________ A X D C C E Y B Use the given information to mark the diagram and any additional congruence you can determine from the diagram. Then complete the triangle congruence statement and give the reason for triangle congruence . B III. 1. C 1 2. A 2 4 D B 3 B A Given: ∠1 ≅ ∠3, ∠2 ≅ ∠4 Given: ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷, ∆𝐴𝐵𝐶 ≅ ∆__________ by __________ 3. C D F ∆𝐴𝐵𝐷 ≅ ∆__________ by __________ A 4. C 1 A 2 G E B ∠𝐴𝐷𝐵 ≅ ∠𝐶𝐷𝐵 D 4 3 B ̅̅̅̅ ̅̅̅̅ 𝑎𝑛𝑑 𝐸𝐴 Given: 𝐺 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐹𝐵 Given: ∠1 ≅ ∠3, ̅̅̅̅ ≅ 𝐴𝐵 ̅̅̅̅ 𝐶𝐷 ∆𝐴𝐵𝐺 ≅ ∆__________ by __________ ∆𝐴𝐵𝐶 ≅ ∆__________ by __________ 41 Congruent Triangles Activity EXAMPLE: 1. Name the congruent triangles Δ_ABC_≅Δ_DEF_ 2. The triangles are congruent by the _ASA_ postulate ∡𝑪 ≅ ∡𝑭 5. Draw the congruent triangles A 3. List all of the congruent, corresponding sides ̅̅̅̅ ≅ ̅̅̅̅ 𝑨𝑩 𝑫𝑬 ̅̅̅̅ ≅ 𝑬𝑭 ̅̅̅̅ 𝑩𝑪 ̅̅̅̅ ≅ 𝑫𝑭 ̅̅̅̅ 𝑨𝑪 4. List all of the congruent, corresponding angles ∡𝑨 ≅ ∡𝑫 ∡𝑩 ≅ ∡𝑬 C D F B E 1. Name the congruent triangles Δ_OSU_≅Δ_BCK_ 2. The triangles are congruent by the _____ postulate 3. List all of the congruent, corresponding sides 5. Draw the congruent triangles 4. List all of the congruent, corresponding angles 1. Name the congruent triangles Δ_____≅Δ_____ 2. The triangles are congruent by the _SAS_ postulate 3. List all of the congruent, corresponding sides 5. Draw the congruent triangles 4. List all of the congruent, corresponding angles 42 1. Name the congruent triangles Δ_____≅Δ_____ 2. The triangles are congruent by the _____ postulate 3. List all of the congruent, corresponding sides ̅̅̅̅̅ ≅ 𝒁𝑰 ̅̅̅ 𝑴𝑹 ̅̅̅̅̅ ̅̅̅̅ 𝑳𝑴 ≅ 𝑴𝑰 ̅̅̅̅ 𝑳𝑹 ≅ ̅̅̅̅̅ 𝑴𝒁 4. List all of the congruent, corresponding angles 5. Draw the congruent triangles 1. Name the congruent triangles Δ_____≅Δ_____ 2. The triangles are congruent by the _____ postulate 3. List all of the congruent, corresponding sides ∡𝑬 ≅ ∡𝑫 5. Draw the congruent triangles 4. List all of the congruent, corresponding angles ∡𝑸 ≅ ∡𝑨 ∡𝑾 ≅ ∡𝑺 1. Name the congruent triangles Δ_____≅Δ_____ 2. The triangles are congruent by the _____ postulate 3. List all of the congruent, corresponding sides 4. List all of the congruent, corresponding angles 5. Draw the congruent triangles F I A X L 43 Day 9 (Quiz Day) : Triangle Congruence Cont. For questions #1-3, use the congruent pentagons below K S E C 4 cm B 5. B corresponds to _____ 6. BLACK ≅ __________ 5 cm R 7. KB = ______ cm A L H ̅̅̅̅ ⊥ 𝐿𝐴 ̅̅̅̅, name two right angles 8. If 𝐶𝐴 4.5 cm O from the figures __________ 9. If ΔDEF ≅ ΔRST, 𝑚∡𝐷 = 100 and 𝑚∡𝐹 = 40, name four congruent angles. _______________________ I. Δ𝑃𝑄𝑅 ≅ Δ𝐴𝐵𝐶. Find the values of x and y. 1. 𝑚∠𝑅 = 5𝑥 + 70, 𝑚∠𝐶 = 24𝑥 − 25, 𝑄𝑅 = 4𝑦 + 2, 𝐵𝐶 = 6𝑦 − 8 2. 𝑚∠𝑅 = 90 − 𝑦, 𝑚∠𝐶 = 13, 𝑃𝑅 = 5𝑥 − 10 , 𝐴𝐶 = 32 − 𝑥 3. 𝑃𝑄 = 5𝑥 − 31, 𝐴𝐵 = −3𝑥 + 1 , 𝑄𝑅 = −3𝑦 − 1, 𝐵𝐶 = −2𝑦 − 2 4. 𝑚∠𝐴 = 15𝑦 − 3, 𝑚∠𝑃 = 12𝑦 + 30, 𝑃𝑄 = 11 − 𝑥, 𝐴𝐵 = 11𝑥 − 1 44 5. 𝐴𝐵 = 2𝑥, 𝑃𝑄 = 18, 𝐵𝐶 = 11, 𝑄𝑅 = 4𝑦 + 3 6. Δ𝑋𝑌𝑍 ≅ Δ𝑀𝑁𝑂, 𝑚∠𝑋 = 𝑥 + 10, 𝑚∠𝑀 = 4𝑥 − 17, 𝑚∠𝑌 = 2𝑦, and 𝑚∠𝑁 = 𝑦 + 6. Find 𝑚∠𝑋 and 𝑚∠𝑌. II. Solve. 1. The perimeter of ABCD is 85. Find the value of x. Is Δ𝐴𝐵𝐶 congruent to Δ𝐴𝐷𝐶? Explain. C 4x 6x - 11 B D 5x - 7 3x + 4 A 45 Day 10: Flow Chart Proofs 46 E is the midpoint of ̅̅̅̅ 𝐵𝐷 47 Use a Flow Chart to prove each of the following triangles congruent. 1. Given BAD Prove ABD BCD CBD 2. Given: B E 3. Given: AC DC C is the midpoint of BE & DA Prove: CAB CDB Prove: ABC DEC 4. Given: AD BC BA CD Prove: ABC CDA 5. Why is this situation not possible to prove? 6. Given BC EF CA FD Prove: BCA EDF 48 1) 2) 3) 4) 49 4) 6) 50 Day 11: Flow Proof Charts Cont. Class work Prove: ADB CDB 51 52 Day 12 – Unit 3 Study Guide: Triangles For # 1-3, describe each triangle by sides and angles. Use the word bank provided. Acute ∆ , right ∆ , Obtuse ∆ , scalene ∆ , isosceles ∆ , equilateral ∆ 1. 2. a. ________________ a. ________________ b. ________________ b. ________________ 3. a.______________________ b. ______________________ For # 4-6, find the given variable(s) for each of the pictures below. 4. t= ___ 5. a = ___ 55 12t+8 72 8t-3 4a+12 52 53 6. y = ___ 30 7y – 10 y 7. Solve for x, y, and z. 32 x= _______ y = ________ z = _______ 2z - 27𝑜 x-6 y + 13 z + 15𝑜 x + 10 For # 8 – 13, state whether the triangles are congruent by : SSS, SAS, ASA, AAS, or not possible. 8. 9. 10. 54 Name the congruent angles and sides for each pair of congruent triangles. 14. Given Δ ABC ≅ ΔDEF ____________ _____________ ____________ ____________ _____________ _____________ For #15-19, find the value of x. 15. 16. 2x - 11 15 M 3x+12 x = ______ 17. x + 48 x = ______ 18. x 4x-12 46 2x + 30 x = ______ x = ______ 19. 80 45 50 70 x = ______ x ***All tests are cumulative! Don’t forget to review Unit 1 material on Transformations! 55 56