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8 Mathematics Quarter 3 - Module 4 TRIANGLE CONGRUENCE Government Property NOT FOR SALE What I Know Pre Assessment: Multiple Choices Directions: Read and answer the questions. Select the letter of the best answer from among the given choices. _______1. Which of the following states the ASA Congruence Postulate? A. The three angles of one triangle are congruent to the three angles of second triangle. B. Two angles and the included side of one triangle is congruent to two angles and the included side of the second triangle. C. Two sides and an included angle of one triangle is congruent to two sides and an included angle of the second triangle. D. Two angles and non-included side of one triangle is congruent to two angles and non-included side of the second triangle _______2. Based on the given illustration below, what congruent postulate/theorem proves the two triangles are congruent? A. ASA B. AAS C. SAS D. SSS ______3. Consider the figures below, what other pair of congruent angles need to be marked so that triangles are congruent by AAS? A. ∠BAE≅∠BCE B. ∠ABE≅∠CBE C. ∠BEA≅∠BEC D. ∠AEB≅∠CEB _______4. The following are pair of congruent triangles EXCEPT A. C. B. D. _______5.The following are right triangle congruent theorems EXCEPT A. If the legs of one right triangle are congruent to the legs of another right triangle, the triangles are congruent. B. If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, the triangles are congruent. C. If the angles of one right triangle are congruent to the angles of the other right triangle, the triangles are congruent. D. If the sides opposite the acute angles of one right triangle are congruent to the sides opposite the acute angles of the second right triangle, the triangles are congruent. _______6. Based on the given illustration at the right, what congruent theorem proves the two right triangles are congruent? A. LL B. LA C. AA D. SS need to be marked so that triangles are congruent by LL Congruent Theorem? A. AC≅DC B. BC≅EC C. AB≅DE D. AD≅BE _______8. The following are pair of congruent triangles EXCEPT A. B. C. D. _______9. What is the value of angle A? a. 450 b. 460 c. 480 d. 500 ________10. If BA= x+2, ED=4, AC=2y and FD=8. Find the value of x and y. a. x=2,y=4 b. x=2,y=2 c. x=4,y=2 d. x=4,y=4 Lesson 1 Proving Two Triangles By SSS,SAS, ASA Congruence Postulate and AAS Congruence Theorem What’s In You have learned from your previous lesson on how to prove congruent triangles using corresponding parts of a triangle. Now, Let us first answer the problem below as a review. A. Name all the corresponding sides and angles below if the polygons are congruent. The first pair is done for you. ̅̅̅̅ ↔ 𝐽𝐾 ̅̅̅̅̅ 𝐴𝐶 ∠A ↔ ∠J ______________ _____________ ______________ ______________ 2 In order to say that two triangles are congruent, you must show that all six pairs of corresponding parts of the two triangles are congruent. However, it is not always necessary to show all the six pairs of congruent parts to prove that the triangles are congruent. These are the postulates and theorem that guarantee the congruence of two triangles by showing only three pairs of congruent corresponding parts. Now, let us see how we can verify if two triangles using two or three pairs of congruent corresponding parts. Activity1: Tell Me Using the given postulate, tell which parts of the pair of triangles should be shown congruent. 1. SAS Side : FE ≅ CD Included Angle: ∠E ≅ ∠D Side: AE ≅ BD Note: ∠E and ∠D are included angles, so you cannot use the sides FA and CB. 2. ASA 3. SSS 4. ASA 5. AAS 3 Identifying Congruent Triangles 4 Example1. Problem: Juan is planning to build a house with triangular roof structure. He wants to make this sketch perfect and accurate in terms of its measurement. He noticed that the roof structure is made up of two triangles. To say that the triangular structure must be perfect and accurate these triangles must have the same size and shape. Can we say that the triangles are congruent based on the sketch? Let A, B and D be the vertices of the first triangle. And C, B and D be the vertices of the second triangle. Observe that if we separate the triangles BD is a common side. Based on the markings, AB≅CB AD≅CD BD≅BD because, BD is common to both triangles. Therefore, ∆ABD≅∆CBD by SSS Congruence Postulate Example 2 From the diagram, you know that BD ≅ CD and AD ≅ AD. The angle included between AD and BD is ∠BDA. The angle included between CD and AD is ∠CDA . Because any two right angles are congruent, ∠ BDA ≅ ∠CDA You can use the SAS Postulate to conclude that ∆𝐴𝐷𝐵 ≅ ∆𝐴𝐷𝐶. 5 Example 3 In the figure, ∠A ≅∠D, ∠B ≅∠E, and AC ≅ PR. The ASA Postulate can be used to show that ∆BAC≅ ∆EDC because AB and DE are included between the congruent angles. Example 4. In the figure, ∠F ≅∠D, ∠E ≅∠B, and ̅̅̅̅ ≅ 𝐵𝐷 ̅̅̅̅. Therefore ∆𝐹𝐴𝐸 ≅ ∆𝐷𝐵𝐶 𝐴𝐹 by AAS Congruence Postulate. What’s More Activity 2: Identify Me Direction: Determine whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write NOT POSSIBLE 1. 2. 3. 4. 5. 6. 6 What I Have Learned Activity 3: The Importance of Triangle 1. List down and draw all structures built in triangles. 2. Can you find parts of these structures where triangles are congruent? Justify your answer 3. Why are bridges stable? https://www.bulbapp.com/u/balsa-wood-bridge~52 What I Can do Activity 4: Picture, Picture Take a picture of objects in your house where you can see triangles. For each picture, identify congruent triangles. Justify why these triangles are congruent by SAS, ASA, SSS and AAS. Do this task in your portfolio. 7 Lesson 2 Congruent Right Triangles What’s In You have proven triangle congruence using SSS Postulate, SAS Postulate, ASA Postulate and AAS Theorem. Now, you will prove right triangles are congruent using the LL, LA, HyL and HyA Congruence Theorem. Activity 1: Let’s Review A. What kind of triangle is shown in the given figure? How can you say that it is a right triangle? a. What is the sum of the angles in a triangle? b. If one is a right angle ( 900), what is the sum of ∠x and ∠y? c. What kind of angle pairs is ∠x and ∠y? d. What type of angles are ∠x and ∠y? e. How do you call the longest side of the triangle? How do you know that it is the longest side? f. How do you call the two other sides of the triangle? g. What are the characteristics of a right triangle? 8 What’s New Activity 2: ART Integration Direction: a. b. c. Draw or sketch two right triangles (b=5 and h=3) Draw or sketch two right triangles with common side (b=5 and h=3) Draw or sketch two right triangles with common vertex. (b=5 and h=3) https://www.sampletemplates.com/business-templates/free-isometric-paper.html How did you draw the two right triangles? How can you say that these right triangles are congruent? _________________________________________________ What Is It Let us consider the tests for proving two right triangles that are congruent. In a right triangle, there is one right angle. The side opposite the right angle is called the hypotenuse. The other two sides are called legs. 9 Consider two right triangles ABC and XYZ such that AB≅ XY and BC ≅YZ. Since all right angles are congruent, then ∠𝐵 ≅ ∠𝑌. Thus, by SAS congruence postulate, we have ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍. . Let us now summarize the above result in the following theorem The LL Congruence Theorem was deduced from SAS Congruence Postulate. Let us now consider right triangles HOT and DAY with right angles at O and A, respectively, such that ∠H≅ ∠D and HT ≅ DY. Since all right angles are congruent, then∠ O≅ ∠𝐴. Thus, by AAS Congruence Theorem, we have ∆𝐻𝑂𝑇 ≅ ∆𝐷𝐴𝑌. We now summarize the above result in the following theorem. The HyA Congruence Theorem was deducted from the AAS Congruence Theorem. Consider now right triangles BIG and SML with right angles at I and M, respectively, such that BI≅SM and ∠B≅ ∠S. Again, since all right angles are congruent, then ∠I≅ ∠M. Thus, by ASA Congruence Postulate, we have ∆BIG≅ ∆SML. We now summarize the above result in the following theorem. 10 The LA Congruence Theorem was deduced from either the ASA Congruence Postulate or the AAS Congruence Theorem. Proof: Let right triangles N and Y with 𝑚∠N = 𝑚∠Y= 900, MK≅XZ and NK ≅YZ. We want to prove that ∆MNK≅ ∆XYZ. Extend the ray XY to point G such that GY ≅ MN and draw GZ. By SAS Congruence Postulate, we have ∆MNK≅XYZ. By CPCTC, we get MK≅GZ. By the transitive property of congruence, we have XZ≅GZ, and hence ∠X≅ ∠G. By AAS Congruence Theorem, it follows that ∆XYZ≅ ∆GYZ. Since ∆MNK≅ ∆GYZ and ∆GYZ≅ ∆XYZ, the transitive property of congruence finally implies that ∆MNK≅ ∆XYZ. 11 What’s More Activity 3: Find my pair In each figure, congruent parts are marked. Give additional congruent parts to prove that the triangles are congruent and state the congruence theorem that justifies your answer. ∠BAD ≅ ∠CAD ∠A BD≅ ∠ACD AB ≅AC AAS Congruence Postulate B. State a congruence theorem on right triangles. 4. 5. 6. 7. 12 Lesson 3 Applying Triangle Congruence What’s In You have learned triangle congruency using postulates and theorems of a triangle. Now, we will apply the triangle congruency in solving problems. What’s New Activity 1: Label and Find! Label the two triangles with the given information and solve for x, y, and z. . 13 You can use the triangle congruence postulates and theorem to solve many real problems that involve congruent triangles. Example 1 Mara bought matching triangular pendants for herself and her sister in the shapes shown. For what the value of x can you use a triangle congruence theorem/postulate that the pendants are congruent? Which triangle congruence theorem can you use? Explain. AB ≅ AC and JK ≅JL, because they have the same measure. So, If CB ≅ KL then ∆𝐶𝐴𝐵 ≅ ∆𝐿𝐽𝐾 by SSS Triangle Congruence Theorem. To solve x, we set 4x-6 = 3x-4 4x-6 = 3x-4 4x = 3x -4 +6 4x-3x = -4 +6 x= 2 Example 2 In the diagram, ∆𝐴𝐵𝐶 ≅ 𝑉𝑇𝑈. Find the indicated measure. a. m ∠B b. AB c. m ∠T d. m ∠V . 14 a. b. 500+700+ m ∠B = 1800 AB = VY 1200 + m ∠B = 1800 AB = 15m m ∠B = 1800-1200= 60 c. d. ∠T ≅ ∠B m ∠T = m∠B = 600 ̅̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝑉𝑇 ∠V ≅ ∠A m ∠V = m ∠A = 500 Example 3 Find the value of x. 1200 + 200 + (2x +4)0=1800 1440 + 2x =1800 2x =1800 -1440 x= 18 Example 4 Find the value of x. 900 + 350 + 5x =1800 1250 + 5x =1800 5x =1800 -1250 x= 11 Activity 2: My X and Y A. Solve the following problems. 1. Find the value of x. a. b. 15 2. Given ∆𝐴𝐵𝐶 ≅ ∆DEF Find the values of x and y. 3. Given ∆𝑈𝐵𝐷 ≅ ∆RAN Find the values of x and y. What I Can Do Activity 3: Fly, Fly, Fly During the Math Fair, one of the activities is a symposium in which the delegates will report on an inquiry about an important concept in Math. You will report on how congruent triangles are applied in real-life. Your query revolves around this situation; 1. Design at most 5 different paper planes using congruent triangles. 2. Let it fly and record the flying time and compare which one is the most stable. 3. Point out the factors that affect the stability of the plane. 4. Explain why such principle works. 5. Draw out conclusion and make recommendations. 16 Procedure: 1. Prepare 5 paper planes 2. Apply your knowledge on triangle congruence. 3. Follow steps 2 to 5. 4. What is the importance of congruent triangles in making paper planes? 17 Summary: In this module you have learned that: • Two triangles are congruent if their vertices can be paired such that corresponding sides are congruent and corresponding angles are congruent. • The three postulates for triangle congruence are: a. SAS Congruence – if two sides and the included angle of one triangle are congruent respectively two sides and the included angle of another triangle then the triangles are congruent. b. ASA Congruence – if two angles and the included side of one triangle are congruent respectively two angles and the included side of another triangle then the triangles are congruent. c. SSS Congruence – if the three sides of one triangle are congruent respectively three sides of another triangles then the triangles are congruent. • AAS Congruence Theorem – if the two angles and the non-included side of one triangle are congruent to the two angles and the non-included side of another triangle than the triangles are congruent. • The congruence theorems for right triangles are: a. LL Congruence – if the legs of one right triangle are congruent respectively to the legs of another right triangle, then the triangles are congruent. b. LA Congruence – if a leg and an acute angle of one triangle are congruent respectively to a leg and an acute angle of another right triangle, then the triangles are congruent. c. HyL Congruence – if the hypotenuse and a leg of one right triangle are congruent respectively to the hypotenuse and a leg of another right triangle, the triangles are congruent. d. HyA Congruence – if the hypotenuse and an acute angle of one right triangle are congruent respectively to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent. 18 Directions: Check which congruence postulate/theorem you would use to prove that two triangles are congruent. 1. 2. 3. 4. 5. 6. 7. 8. 19 9. 10. B. Find the value of x or y so that .∆UBD≅ ∆RAN. 11. 12. 13. Given: ∠U ≅ ∠N 20