Mathematics Quarter 3 – Module 2 (Week 3 & 4) PDF

8 Mathematics Quarter 3 – Module 2 (Week 3 & 4) Triangle Congruence About the Module This module was designed and written with you in mind. It is here to help you master about Triangle Congruence. The scope of this module permits it to be used in many different learning situations. The lan...

8 Mathematics Quarter 3 – Module 2 (Week 3 & 4) Triangle Congruence About the Module This module was designed and written with you in mind. It is here to help you master about Triangle Congruence. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. This module contains: Lesson 1 – Triangle Congruence After going through this module, you are expected to:  illustrate triangle congruence; and  illustrate the SAS, ASA, and SSS congruence postulates. ii  What I Know (Pre-Test) Instructions: Choose the letter of the correct answer. Write your chosen answer on a separate sheet of paper. 1. Two triangles are congruent if and only if all of their corresponding parts are similar. A. False statement C. Always True B. True statement D. Sometimes True 2. In a triangle, when the vertices of the two angles are the endpoints of the segment, the segment is said to be the ______________ of the two angles. A. included angle C. included triangle B. included side D. none of these 3. What congruence postulate states that “If three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent”? A. ASA Congruence Postulate C. SSS Congruence Postulate B. SAS Congruence Postulate D. LL Congruence Theorem 4. By inspection, which of the following figures guarantees that the two triangles are congruent by Hypotenuse-Angle Theorem? A. C. B. D. Use the figures below to answer Nos. 5 – 6. 5. If DBA  FEG , which angle corresponds to EGF ? A. BCD C. BAD B. CDB D. CBD 6. Suppose the three triangles are congruent, which of the following statements of congruence is NOT TRUE? A. EFG  BCD C. GFE  ADB B. ABD  CBD D. BDA  BDC 1 7. What congruence theorem can be used to prove that the triangles on the right are congruent? A. LA Congruence Theorem B. LL Congruence Theorem C. HyA Congruence Theorem D. HyL Congruence Theorem 8. A student is trying to prove ABD to be congruent to CBD. She notices the common side, BD  BD and realizes that the simplest way to prove congruency using this information is by: A. ASA B. SAS C. SAA D. SSS 9. If MAN  BOY , then A  ____. A. B B. N C. O D. Y 10. If PQR  XYZ , then QP  _____. A. PQ B. QR C. YZ D. YX 2  Lesson TRIANGLE CONGRUENCE 1 What I Need To Know At the end of this lesson, you are expected to: o define and illustrate the idea of congruence in real-life; o illustrate triangle congruence; o illustrate the SAS, ASA, and SSS congruence postulates; and o illustrate the LL, LA, HyA and HyL congruence theorems. What’s In In almost all places, designs and patterns having the same size and the same shape abound. They are seen in twin towers, skyscrapers, bridges, condominiums, in furniture and appliances, and even in fabrics and handicrafts. Congruence of triangles is studied because of its many applications in the real world. Triangles are considered to be the most stable of all geometric figures, these are oftentimes used as frameworks, supports, or braces for many construction works. In this module, you will discover the postulates in determining the congruency between triangles. What’s New You must be well aware of a triangle by now — that it is a figure with three sides, three angles, and three vertices. Two or more triangles are said to be congruent if and only if their corresponding sides and angles are congruent. Congruence means having the same shape and size, and it is denoted by ≅. The top part of the symbol “ ~ ” is the sign for similarity and indicates the same shape. The bottom part “ = ” is the sign of equality and indicates the same size. The symbol “↔” denotes correspondence and is read as “corresponds to”. 3  Consider the two triangles on the right ∆ABC and ∆DEF, we can identify the corresponding parts as follows: VERTICES ANGLES SIDES A↔D A  D AB ↔DE B↔E B  E BC ↔ EF C↔F C  F AC ↔ DF Thus, ∆ABC ↔ ∆DEF. ILLUSTRATIVE EXAMPLE 1: Write the correspondence between the two figures. Answer: VERTICES ANGLES SIDES P↔L P  L PR ↔LK Q↔J Q  J PQ ↔ LJ R↔K R  K QR ↔ JK In a triangle, when the vertices of the two angles are the endpoints of the segment, the segment is said to be the included side of the two angles. When the sides of an angle are the two sides of the triangle, then the angle is said to be the included angle of the two sides. ILLUSTRATIVE EXAMPLE 2: Identify the included side of R and P and the included angle of RP and RQ. Answer: The included side of R and P is RP. The included angle of RP and RQ is R. 4 ILLUSTRATIVE EXAMPLE 3: Identify the following in ABC. 1. Included side of C and B 2. Included angle of AB and AC 3. Opposite angle of BC 4. Opposite side of A Answer: 1. Included side of C and B is CB. 2. Included angle of AB and AC is A. 3. Opposite angle of BC is A. 4. Opposite side of A is BC. What is It In order to say that the 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. The postulates that you will discover guarantee that two triangles are congruent given only three pairs of corresponding parts. SIDE-ANGLE-SIDE (SAS) CONGRUENCE POSTULATE If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. If MP  RS , P  S , and PN  SQ , then MPN  RSQ by SAS Congruence Postulate. ANGLE-SIDE-ANGLE (ASA) CONGRUENCE POSTULATE If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. If B  E , BC  EF , and C  F , then ET by ASA Congruence Postulate. 5  SIDE-SIDE-SIDE (SSS) CONGRUENCE POSTULATE If three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. If AB  DE , AC  DF and BC  EF then ABC  DEF by SSS Congruence Postulate. Right triangles are special because these have right angles and right angles are congruent. Below are the congruence theorems that will help you identify if two right triangles are congruent. LEG-LEG (LL) CONGRUENCE THEOREM If the legs of one right triangle are congruent to the legs of another right triangle, then the two triangles are congruent. If AB  DE , BC  EF and B and E are right angles, then ABC  DEF. LEG-ANGLE (LA) CONGRUENCE THEOREM If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, then the two triangles are congruent. If BC  EF , C and  F are acute angles, B and E are right angles, then ABC  DEF. HYPOTENUSE-ANGLE (HyA) CONGRUENCE THEOREM If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent. If AC  FD , A and  F are acute angles, B and E are right angles, then ABC  DEF. 6  HYPOTENUSE-LEG (HyL) CONGRUENCE THEOREM If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent. If AB  DE , CB  FE and C and  F are right angles, then ACB  DFE. What’s More NOW IT’S YOUR TURN! Activity 1.1 A. Name the required side or angle 1. the included side of M and MEP _____________________________ P 2. the included angle of ET and EN N _____________________________ M E 3. the angle opposite PM _____________________________ T 4. the side opposite N _____________________________ B. State whether the triangles are congruent or not. If the triangles are congruent, write a congruence statement and name the postulate that guarantees congruency between each pair of triangles. 1. 4. 2. M 5. P 3. N O R 7 C. State the theorem that proves that the pair of triangles are congruent. 1. 4. 2. 5. 3. What I Need to Remember  Two triangles are congruent if and only if their corresponding sides and angles are congruent.  TRIANGLE CONGRUENCE POSTULATES  SAS Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.  ASA Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.  SSS Congruence Postulate If three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. 8   RIGHT TRIANGLE CONGRUENCE THEOREMS  LEG-LEG (LL) CONGRUENCE THEOREM If the legs of one right triangle are congruent to the legs of another right triangle, then the two triangles are congruent.  LEG-ANGLE (LA) CONGRUENCE THEOREM If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, then the two triangles are congruent.  HYPOTENUSE-ANGLE (HyA) CONGRUENCE THEOREM If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.  HYPOTENUSE-LEG (HyL) CONGRUENCE THEOREM If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent. What I Can Do Is it possible to prove that the two triangles are congruent by showing three angles of one triangle are congruent to the corresponding three angles of another triangle? In other words, is Angle-Angle-Angle (AAA) a test for a triangle congruence? DO THIS !!! Use a ruler and a protractor to draw a triangle with angles 40°, 60°, 80°. Draw another triangle with the same angle measurements as the first triangle. Are the two triangles congruent? Or do you get two different triangles? So, what do you conclude? Is AAA a test for triangle congruence? 9. Assessment (Post Test) Instructions: Choose the letter of the correct answer. Write your chosen answer on a separate sheet of paper. 1. Two triangles are congruent if and only if all of their corresponding parts are congruent. A. False statement C. Always false B. True statement D. Sometimes false 2. When the sides of an angle are the two sides of the triangle, then the angle is said to be the ____________ of the sides. A. included angle C. included triangle B. included side D. none of these 3. What congruence postulate states that “If two angles and the included side on one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent”? A. ASA Congruence Postulate C. SSS Congruence Postulate B. SAS Congruence Postulate D. LL Congruence Theorem 4. By inspection, which of the following figures guarantees that the two triangles are congruent by Leg-Leg Congruence Theorem? A. C. B. D. Use the figures below to answer Nos. 5 – 6. 5. If DBA  FEG , which side corresponds to FG ? A. AC B. BD C. AD D. DC 6. Suppose the three triangles are congruent, which of the following statements of congruence is NOT TRUE? A. ABD  CBD C. GFE  ADB B. EFG  BCD D. BDA  BDC 10 7. What congruence theorem can be used to prove that the triangles on the right are congruent? A. LA Congruence Theorem B. LL Congruence Theorem C. HyA Congruence Theorem D. HyL Congruence Theorem 8. Which of the following statements is FALSE? A. If the legs of one right triangle are congruent to the legs of another right triangle, then the two triangles are congruent. B. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. C. If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent. D. If two angles and the included side of one triangle are congruent to hypotenuse and corresponding leg of another triangle, then the two triangles are congruent. 9. If MAN  BOY , then Y  ____. A. B B. N C. O D. Y 10. If PQR  XYZ , then QP is the opposite of _____. A. P B. Q C. R D. XY 11 Images: https://images.app.goo.gl/SGUfptgCRgykDtpr9 https://images.app.goo.gl/ZwewqKb3554eqyYM6 https://images.app.goo.gl/V8LUyVvZZ8FEa1oJ7 https://images.app.goo.gl/6QzggVu4Cq7ycVVYA https://images.app.goo.gl/jfTVRm1fo66CZ5Pe6 https://images.app.goo.gl/GqmjNnY6m9EtqPqKA https://images.app.goo.gl/VMfjeKLsNBFwVtMq8 https://images.app.goo.gl/eSC5T4aeru9mGWMUA https://images.app.goo.gl/QUjPvPJ3uC8VNSiL7 https://images.app.goo.gl/sUWEqqpDBmbGoGLN8 https://images.app.goo.gl/RfeUdDe376fnn6pu6 https://images.app.goo.gl/T2pd8GvDdAQ3odAc8 https://images.app.goo.gl/Pj2XQUcrKyQ1HFRh7 Congratulations! You are now ready for the next module. Always remember the following: 1. Make sure every answer sheet has your o Name o Grade and Section o Title of the Activity or Activity No. 2. Follow the date of submission as agreed with your teacher. 3. Keep the modules with you. 4. Return them at the end of the school year. 13