Module 2 similarity
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Here are the key steps: 1. ∆MAN ~ ∆MON (by AAA similarity theorem) 2. There is 1 triangle similar to ∆MAN For the polyominoes activity: - A polyomino made of 1 square would require 4 sticks - A polyomino made of 2 squares would require 6 sticks - A polyomino made of 3 squares would require 8 sticks - A polyomino made of 4 squares would require 10 sticks (as in the example given) - Continuing the pattern, a polyomino made of n squares would require 2n + 2 sticks For the rectangle counting activity: - There are 4 rectangles in the given diagram (
Module 2 similarity
- 1. Module 17 Similar Triangles What this module is about? This module is about similar triangles, definition and similarity theorems. As you go over the exercises you will develop your skills in determining if two triangles are similar and finding the length of a side or measure of an angle of a triangle. What are you expected to learn? This module is designed for you to: 1. apply the definition of similar triangles in: a. determining if two triangles are similar b. finding the length of a side or measure of an angle of a triangle 2. verify the Similarity Theorems: a. AAA Similarity b. SAS Similarity c. SSS Similarity 3. apply the properties of similar triangles and the proportionality theorems to calculate lengths of certain line segments. How much do you know? A. For each pair of triangles, indicate whether the two triangles are similar or not. If they are similar, state the similarity theorem or definition that supports your answer. 1. 2 4 450 3 450 6
- 2. 2. 3 1 3 2 3. 3 2 2 3 4. 800 500 500 5. 6 4 4 3 2 12 B. 6 – 10. The length of the sides of a triangle are 14, 8 and 6. Find the length of the two sides of a similar triangle if the length of the shortest side is 12.
- 3. Lesson 1 Definition of Similar Triangles Similar Triangles Two triangles are similar if corresponding angles are congruent and corresponding sides are proportional. Y Examples: 1. If in ∆ ABC and ∆ XYZ ∠A≅ ∠X ∠B≅ ∠Y ∠C≅ ∠Z B A C X BC AC AB = = XY YZ XZ Z then ∆ ABC ∼ ∆ XYZ (Triangle ABC is similar to triangle XYZ) 2. ∆ BMP ∼ ∆ SEC Find the value of x and y E M 3 B 4 5 P S Since the corresponding sides are proportional BM MP BP = = SE EC SC 5 4 3 = = x 8 y 1 5 3 = = x 2 y 1 1 5 3 = ; = x 2 2 y x = 6 y = 10 8 x y C
- 4. Exercises: A. Given: ∆MOL ∼ ∆REY Fill the blanks. 1) ∠M ≅ ______ 2) ∠O ≅ ______ 3) _____ ≅ ∠Y MO ML 4) = RY ? 5) 15 2 5 2 3 L 4 O Y 6 2 ? = 3 15 7) R ML LO = RY ? 6) M 4 2 = 6 ? B. State whether the proportion is correct for the indicated similar triangles 1. ∆RST ∼ ∆XYZ 4. ∆DEF ~ ∆HIS RS ST = XY YZ DE EF = HI IJ 2. ∆ABC ~ ∆DEF 5. ∆KLM ~ ∆PQR AB BC = DE EF KM LM = PR QR 3. ∆RST ~ ∆LMK 6. ∆XYZ ~ ∆UVW ST RT = LM MK XY XZ = UV UW E
- 5. Complete the proportions 7. ∆ABC ~ ∆DEF 9. ∆RST ~ ∆XYZ AB BC AC = = ? ? ? XY XZ YZ = = ? ? ? 8. ∆KLM ~ ∆RST 10. ∆MNO ~ ∆VWX ? ? ? = = KL LM KM VX VW WX = = ? ? ? C. Find the missing length 1. ∆ABC ~ ∆XYZ find b and a Y B 3 a A b 5 10 C X Z 15 2. ∆SIM ~ ∆PON SI = 6cm. IM = 4cm. SM= 8cm, Find the lengths of the sides of ∆PON if the ratio of the lengths of the corresponding sides is 1:3. M 8cm N 4cm. S 6cm. I P O
- 6. Lesson 2 AAA Similarity Theorem If in two triangles the corresponding angles are congruent, then the two triangles are similar. O If ∆BOS ↔ ∆VIC ∠B ≅ ∠V , ∠O ≅ ∠I , ∠S ≅ ∠C then ∆BOS ~ ∆VIC I B S V C AA Similarity If two angles of one triangle are congruent to the corresponding two angles of another triangle, the triangles are similar. A Given: ∠A ≅ ∠O O ∠J ≅ ∠B Then: ∆JAM ~ ∆BON J B M N Examples: Are the triangles similar by AA Similarity? 1. 330 1000 330 470 470 1000
- 8. SAS Similarity Theorem If in two triangles two pairs of corresponding sides are proportional and the included angles are congruent, then the triangles are similar. C O If ∆ICE ↔ ∆BOX CI CE = OB OX and ∠C ≅ ∠O then ∆ICE ~ ∆BOX B I E Examples: A C 9 B ∆AOB ~ ∆DOC by SAS AO BO since = and DO CO ∠AOB ≅ ∠DOC 8 6 O 12 D R Can you explain why is ∆RAT ~ ∆RAM by SAS ? 4 T M 3 A 3 X
- 9. SSS Similarity Theorem If the two triangles three corresponding sides are proportional, then the triangles are similar. L U If ∆SUN ↔ ∆BLK SU UN SN = = BL LK BK then ∆SUN ~ ∆BLK Example: 1. S N B O SN NA SA = = OB PE DE 4 6 8 = = 6 9 12 S 6 4 8 6 N P since the corresponding sides are proportional then ∆POE ~ ∆NSA by SSS Similarity. 12 9 A E 2. Explain: Any two congruent triangles are similar. K
- 10. Exercises: A. Tell whether the two triangles are similar. Cite the Similarity Postulate or Theorem to justify your answer. (Identical marks indicate ≅ parts.) 1. 6 8 3 10 5 2. 6 3 8 4 3. 3 6 6 3 4. 9 4 8 6 10 15 12
- 12. R 9. QS RS = PS ST Q S P T 10. B. Determine whether each pair of the following triangles are similar by SAS, AAA, SSS or not at all. 1. 750 14 12 7 750 6
- 14. C. Answer the following: 1. Two isosceles triangles have an angle of 500. Does it follows that the triangles are similar? 2. Two angles of ∆BEL have measures of 20 and 50. Two angles of ∆JAY have measures of 30 and 100. IS ∆BEL ~ ∆JAY ? 3. Is it possible for two triangles to be similar if two angles of one have measures 50 and 75, where as two angles of the other have measures 55 and 70? 4. Two angles of have measures 40 and 80, where as the two angles of the other have measures 60 and 80, are the two triangles similar? 5 – 6. The lengths of the sides of a triangle are 12 and 15. If the length of the shortest side of a similar triangle is 12, find the lengths of the other two sides. E 7- 8. In the figure, if AE = 8 AB=4, BC=10, ED=3 Find BD and DC. A D 500 500 B 9 – 10. Explain: Any two equilateral triangle are similar. C
- 15. Lesson 3 Application of the Properties of Similar Triangles to Calculate Lengths of certain line segments. Examples: ∆ABC ~ ∆DEF . Find the missing measure. 1. AB = 36, BC = 24, DE = 48 m∠B = 110, m∠E = 110; EF =_______. AB BC = DE EF 36 24 = 48 x 36x = 1152 x = 32 EF = 32 2. AB = 38, BC =24, AC=30, DE=12, EF=16, DF= _______. AB AC = DE DF 18 30 = 12 x 3 30 = 2 x 3x = 60 x = 20 DF = 20 3. AC = 15, DE=12, DF=20, m∠A = m∠D = 35 , AB =__________ AB AC = DE DF x 15 = 12 20 x 3 = 12 4 4x = 36 x=9 AB = 9
- 16. 4. The lengths of the sides of a triangle are 14, 8 and 6. Find the perimeter of a similar triangle if the length of its shortest side is 12. Let x = the length of one side of a triangle x = the length of the longest side of the triangle 6 8 = 12 x 1 8 = 2 x x = 16 6 14 = 12 y 1 14 = 2 y y = 28 P = 12 + 16 + 28 P = 56 5. On a level ground, a 5ft. person and a flagpole cast shadows of 10 feet and 60 feet respectively. What is the height of the flagpole? Let x = the height of the flagpole 5 10 = x 60 5 1 = x 6 x = 30 feet the height of the flagpole
- 17. Exercises 1. A yardstick casts a shadow of 24in. at the same time an electric post cast a shadow of 20ft. 8 in. What is the height of the electric post? 2. Two triangles are similar. The lengths of the sides of one triangle are 5, 12 and 13. Find the lengths of the missing sides of the other triangle if its longest side is 39. 3. The perimeter of a triangle is 32cm. and the ratio of the sides is 3:6:7. Find the length of each side of the triangle. 4. A tall building at Makati casts a shadow of 12m. at the same time a 7m. light pole cast a shadow of 3m. Find the height of the building. 5. If the shadow of the tree is 20m. long and the shadow of the person, who is 190cm. tall, is 250cm. long. How tall is the tree? Let’s Summarize 1. Two triangles are similar if their vertices can be paired so that corresponding angles are congruent and the lengths of corresponding sides are proportional. 2. The AAA Similarity. If the corresponding of two triangles are congruent, then the two triangles are similar. 3. The AA Similarity. If two pairs of corresponding angles of two triangles are congruent, then the two triangles are similar. 4. The SSS Similarity. If the lengths of corresponding sides of two triangles are proportional, then the two triangles are similar. 5. The SAS Similarity. If one pair of corresponding angles of two triangles are congruent and the lengths of the corresponding sides that include these angles are proportional, then the two triangles are similar.
- 18. What have you learned? A. Given the figures and the information below can you conclude that ∆KMN ~ ∆RPS ? If so what Similarity Theorem? M 1. m∠M = 80 m∠P = 80 16 R 24 K 24 36 2. m∠M = 80 , m∠R + m∠S = 100 P S 3. RS = 26 KN = 40 N B. Find the missing side 20 4. 24 16 x 5 6 5. 3 4 5 9 12 X =? 6 – 7. If the shadow of the tree is 14cm. long and the shadow of the person who is 1.8m. tall is 4m long, how tall is the tree? 8 – 10. A pole 3m. high has a shadow 5m long when the shadow of a nearby building is 110m. long. How tall is the building?
- 19. Key How much do you know? 1. 2. 3. 4. 5. 6. yes, SAS No yes, SAS yes, AAA yes, SSS 16,28 Lesson 1 Exercises: A.1. ∠R 2. ∠E 3. ∠L 4. RE 5. YE 6. 5 7. 3 Lesson 2 A.1. Similar, SSS 2. Similar, SAS 3. Similar, SAS 4. Similar, SSS 5. Similar, AAA 6. Similar, SSS 7. Similar, SAS 8. not similar 9. Similar, SAS 10.Similar, SSS B. 1. Correct 2. correct 3. Not correct 4. correct 5. correct 6. correct 7. DE, EF, DF 8. RS, ST, RT 9. RS, RT, ST 10. MO, MN, NO C. 1. a=6 b=1 B.1. Similar, SAS 2. Similar, AA 3. Similar, SAS, AA 4. Similar, AA 5. Similar, SSS C.1. 2. 3. 4. 5 6 7. 8. 9 10 2. NP=24 NO=12 PO = 18 yes no no yes 18, 22.5 BD = 5.7’ DC = 7.5 Equilateral ∆ is equiangular by AAA. Any 2 equilateral ∆ are similar.
- 20. Lesson 3 Exercises 1. 2. 3. 4. 5. 372in. 15, 36 6cm., 12cm., 28m. 152cm. 14cm. What have your learned? 1. Similar, SAS 2. Similar, SAS 3. not similar 4. 4 5. 15 6-7. 6.3m 8 - 10. 66m.
- 21. Fun with Math Activities Related to Similar Figure
- 22. A B E C D O M L D List all the triangles in the figure that are similar to ∆MAN . N
- 23. Polymonies are made up of a number of squares connected by common sides. Thirteen sticks were used to make this one with four squares. others. Investigate the numbers of sticks needed to make
- 24. How many rectangles are there in this diagram?
- 25. Make a rectangular grid of squares. Cut it along the grid lines to make two identical pieces.