Examples: 1. Write the angles in order from 2. Write the sides in order from


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1 Lesson 1 Triangle Inequalities 17. I can apply the triangle inequalities theorems When considering triangles, two basic questions arise: Can any three sides form a triangle? What is the relationship between the angles and sides of a triangle? NO show how sides of 4, 6, and 1 can t SHORTEST SIDE ACROSS FROM SMALLEST ANGLE, LONGEST SIDE ACROSS FROM LARGEST ANGLE Examples: 1. Write the angles in order from. Write the sides in order from smallest to largest shortest to longest F, H, G PQ, QR, PR 3. Determine whether a triangle can have the given side lengths. Show work or explain your reasoning. 7, 10, 19.3, 3.1, 4.6 8, 13, 1 NO YES NO ( = does not count!) 4. A triangle has side lengths 7 and 1. What is the range of possible side lengths for the other side? 1 7 5, x 19 Third side is between 5 and 19
2
3 Lesson Ratios and Proportions 1. I can express ratios in multiple formats. I can solve proportions A ratio compares two numbers by division. It can be written in three ways: 10 : to 1 A proportion is an equation stating two ratios are equal. a b c d Cross Products Property: In a proportion, a c a d b c b d Other Properties of Ratios The proportion a c b d a b is equivalent to: * * d c c d b a * b d a c Examples: 1. Solve the following proportions: 7 56 a. x 7 56x 504 x 9 b. x 3 x 5 4 5x 4(x3) 5x 8x1 3x 1 x 4 c. x x 4 d. x 3 8 x 3 ( x 4) 100 x 4 10 x 14 or x 6 ( x 3)( x 3) 16 x x x 5
4 . Given that 6 5 x y, then x y =? x y According to a recent study, 5 out of 6 high school students have a smart phone. If there are 900 students at Gull Lake High School, approximately how many of them have a smart phone? 5 x 6x 4500 x students Let x be the length of the model in centimeters. The rectangular model of the racing car is similar to the rectangular racing car, so the corresponding lengths are proportional. Find the length of the model to the nearest tenth of a centimeter. actual x 31.5 x 17.5 model 6.3 x Length of model is 17.5 cm 5. The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of the shortest side? 4x 7x 5x 96 16x 96 x 6 5(6) 30 Shortest side is 30 cm
5 Lesson 3 Similarity in Figures 3. I can state the properties of similarity 4. I can find the similarity ratio of similar triangles 6. I can find missing angles of similar triangles 10. I can write a similarity statement 11. I can verify that triangles are similar We have already learned what it means for two figures to be congruent. We are going to see what it means for two figures to be similar. Two figures are congruent if and only if they have the same shape and size. If two figures are congruent, then their corresponding angles are congruent and their corresponding side lengths are congruent. Congruence Transformations: Two figures are similar if and only if they have the same shape, but not necessarily the same size. If two figures are similar, then their corresponding angles are congruent and their corresponding side lengths are proportional. Similarity Transformations: The similarity ratio is ratio of lengths of corresponding sides. The similarity ratio of ABCD to EFGH is 1: The similarity ratio of EFGH to ABCD is :1 When figures are congruent to each other their similarity ratio is 1: 1
6 Examples: Determine if each pair of figures is similar to each other. If so, identify the corresponding sides and angles and determine the similarity ratio and similarity statement. If not, explain why they are not similar. M N L P J S Ratio: LM MJ JL PN NS SP Statement: JLM SPN A E B F C G D H Ratio: 3 Statement: ABCD BC CD DA AB FG GH HE EF EFGH NOT similar, angles are not congruent A J B G C H AB BC AC JG GH JH Ratio: 1 Statement: ABC JGH
7 Lesson 4 Triangle Similarity 5. I can find the missing side lengths of similar triangles 7. I can show triangles are similar using the AA Postulate 8. I can show triangles are similar using the SAS Theorem 9. I can show triangles are similar using the SSS Theorem 10. I can write a similarity statement 11. I can verify that triangles are similar Just like we learned that there are shortcuts to proving triangles congruent to each other, there are shortcuts to proving triangles similar to each other. They are: AA~ SSS~ SAS~ Examples: Prove or explain why the triangles are similar and write a similarity statement. A D (right 's) BCA ECD (vertical 's) ABC DEC by AA~ PR PQ QR SU ST TU PQR STU 3 by SSS~ TX UX VX WX TXU WXV (vertical 's) TUX VWX by SAS~ D H (given) DE FD 5.8 HJ KH DEF HJK by SAS~ B E (right 's) A D (both 43 ) ABC DEF by AA~
8 Refer to the diagram to the right. Explain why ABE ~ ACD. B C (right 's) A A (reflexive) ADC AEB by AA~ Now that you know the triangles are similar, determine CD. small big x 60 9x x 6.6 Refer to the diagram at the right. Explain why ABE ~ ACD. A A (reflexive) ABE ACD (corr. 's) ABE ACD by AA~ Now that you know the triangles are similar, determine BE and CD. small 3 x big 7.5 x 6 3x x x x 4 BE 4 CD 10
9 Lesson 5 Properties of Similar Triangles 1. I can apply the Triangle Proportionality Theorem 13. I can apply the Converse of the Triangle Proportionality Theorem 14. I can apply the TwoTransversal Proportionality Corollary 15. I can apply the Triangle Angle Bisector Theorem Using the theorems involving proportional relationships, you can complete the following examples. Examples: In each example, find what s asked for and NAME the theorem you used. Find PN Find SU SU 4 10 SU 56 SU 5.6 SideSplitter Theorem 3 PN 5 PN 15 PN 7.5 SideSplitter Theorem Given that AC 36, what value of BC would make DE AB? Verify that DE BC Conv. SideSplit Thm BE 0 BE 40 BE 1 BC 7
10 LM 4.5 LM 1.74 LM MN 4.5 MN 0.09 MN 4.5 Find LM and MN Find PS and SR x 3 x x 80 3x 160 8x 40 x 30 PS 8 SR 35 Find AC and CD y y 4y 4.5y 9.5x 9 x 18 AC 16 DC 9
11 Lesson 6 Indirect Measurement 16. I can solve realworld problems using similar triangles Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. The following examples use indirect measurement to find a missing measure. For each of the following, draw a sketch of the situation and use similar triangles to determine the missing length. Using an object s shadow Follow along: Tyler wants to find the height of a telephone pole. He measured the pole s shadow and his own shadow and then made a diagram. What is the height h of the pole? height 5'9" H 69 H 9H H '9" shadow 7'8" 38' 4" You Try: A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole. What is LM? height 5' 6" H 66 H 60H 110 H '7" shadow 5' 14' " Using a scale drawing Follow along: On a Wisconsin road map, Kristin measured a distance of 11 in. from Madison to Wausau. The scale of this map is 1inch: 13 miles. What is the actual distance between Madison and Wausau to the nearest mile? map actual 1 11 x 143 miles 13 x You Try: The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in.:0 ft. drawing 1 L 0L inches long actual 0 74 drawing 1 L 0L 60 3 inches long actual 0 60
12 What is the relationship between Similar Figures, Perimeter, and Area? The following figures are similar squares, similar triangles, and similar rectangles. Find the similarity ratio, and perimeter and area of each figure. A. Similar Squares B. Similar Triangles C. Similar Rectangles P: 8 P: 4 P: 1 P: 4 P: 30 P: 18 A: 4 A: 36 A: 6 A: 4 A: 50 A: 18 Similarity Ratio: 3:1 :1 5:3 Perimeter Ratio: 3:1 :1 5:3 Area Ratio: 9:1 4:1 5:9 Follow along: Given that LMN QRT, find the perimeter P and area A of QRS. a 9.1 b Perimeter: cm Area: cm You try: ABC ~ DEF, BC = 4 mm, and EF = 1 mm. If P = 4 mm and A = 96 mm for DEF, find the perimeter and area of ABC. a 1 b 3 1 Perimeter: 4 14 cm 3 1 Area: cm
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