Mathematics Quarter 3 Module 4 Proving Theorems on the Different Kinds of parallelogram answer Key

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\ Republic of the Philippines Department of Education Regional Office IX, Zamboanga Peninsula 9 Zest for Progress Zeal of Partnership Mathematics Quarter 3 - Module 2: Theorems of a Parallelogram and Its Proof Name of Learner: Grade & Section: 0 Name of School:

In the previous module, you learned about the different properties of parallelograms. In this module you will learn to prove theorems on the different kinds of parallelogram (rectangle, rhombus, square). What I Need to Know Learning Competency: In this module, you will be able to: prove theorems on the different kinds of parallelogram (rectangle, rhombus, square). (M9AL- IIIc-1) What I Know Find out how much you already know about this lesson. Encircle the letter of the correct answer. Take note of the items that you were not able to answer correctly and find out the right answer as you go through this module. 1. What condition will make parallelogram WXYZ a rectangle? a. WWWW YYYY c. X is a right angle b. WWWW YYYY d. WWWW and YYYY bisect each other 2. Which of the statement is true about theorem on rhombus? a. The diagonals of a rhombus are not perpendicular b. The diagonals of a rhombus are parallel c. Each diagonals of a rhombus do not bisect opposite angles d. Each diagonals of a rhombus bisects opposite angles 3. Which of the statements describes that a quadrilateral is a parallelogram? a. Diagonals bisect each other b. The two diagonals are congruent c. Two consecutive sides are congruent d. Two consecutive angles are congruent 4. How many congruent triangles are formed when a diagonal of parallelogram is drawn? a. 1 b. 2 c. 3 d. 4 5. What does CPCTC in mathematics stands for? a. Corresponding parts of congruent triangles are congruent b. Corresponding parts of congruent triangles are coherent c. Corresponding parts of convex triangles are congruent d. Corresponding parts of convex triangles are coherent 1

For numbers 6, 7, and 8: Complete the two-column proof table below. D O Given: DOVE is a rectangle, with diagonals DDDD and EEEE E V Prove: DDDD EEEE STATEMENTS REASONS 1. (6) 1. Given 2. DDDD OOOO 2. (7) 3. DDDDDD and OOOOOO are right angles 3. Theorem no. 1 4. DDDDDD and OOOOOO 4. All right angles are congruent 5. EEEE VVVV 5. Reflexive Property 6. DEV OVE 6. SAS congruence postulate 7. DDDD EEEE 7. (8) 6. 7. 8. a. DOVE is a rectangle, with diagonals DDDD and VVVV b. DOVE is a rectangle, with diagonals DDDD and EEEE c. DOVE is a rectangle, with diagonals DDDD and EEEE d. DOVE is a rectangle, with diagonals DDDD and EEEE a. Opposite sides of a parallelogram are bisects each other b. Opposite sides of a parallelogram are perpendicular c. Opposite sides of a parallelogram are parallel d. Opposite sides of a parallelogram are congruent a. CPCTC b. CPCCT c. SAS congruence postulate d. SSS congruence postulate For numbers 9 and 10: Complete the two-column proof with the following given below. A 7 8 B Given: Rhombus ABCD D 5 6 C Prove: 5 6 7 8 1. Rhombus ABCD 1. Given 2. DDDD AAAA ; BBBB CCCC 2. Definition of Rhombus 2

3. BBBB DDDD 3. Reflexive Property 4. DAB BCD 4. 5. 5. CPCTC 9. Which is the best answer for statement number 5? a. 5 8 ; 7 8 b. 5 7 ; 7 8 c. 5 6 ; 7 8 d. 5 6 ; 7 6 10. Which is the best answer for reason number 4? a. SSA Congruence Postulate b. SAS Congruence Postulate c. ASA Congruence Postulate d. SSS Congruence Postulate What s In Activity 1: Check Your Guess Directions: In the table that follows, write AT in the second column if you guess that the statement is always true, ST if it s sometimes true, and NT if it is never true. You are to revisit the same table later and respond to your guesses by writing R if you were right or W if wrong under the third column. Statement 1. A rectangle is a parallelogram 2. A rhombus is a square 3. A parallelogram is a rectangle 4. A rhombus is a parallelogram 5. A rectangle is a rhombus 6. A square is a rhombus 7. A rhombus is a rectangle 8. A parallelogram is a rhombus 9. A square is a parallelogram 10. A square is a rectangle My guess is (AT, ST or NT) I was (R or W) 3

What s New Activity 2: I Want to Know! Directions: Do the procedures below and answer the questions that follow. Materials Needed: bond paper, protractor, ruler, pencil, and compass Procedure: 1. Mark two points O and P that are 10 cm apart. 2. Draw parallel segments from O and P which are 6 cm each, on the same side of OOOO and are perpendicular to OOOO. 3. Name the endpoints from O and P as H and E, respectively, and draw HHHH. 4. Draw the diagonals of the figure formed. Questions: 1. Measure the OHE and PEH. What did you find? 2. What can you say about the four angles of the figure? 3. Measure the diagonals. What did you find? 4. Does quadrilateral HOPE appear to be a parallelogram? Why? 5. What specific parallelogram does it represent? 4

What Is It Theorem 1: If a parallelogram has a right angle, then it has four right angles and the parallelogram is a rectangle. Given: WINS is a parallelogram with W is a right angle. Prove: II, NN, and SS are right angles W I Proof: S N 1.WINS is a parallelogram with WW is a right 1. Given angle. 2. WW = 90 o 2. Definition of right angle 3. WW NN and II SS 3. In a parallelogram opposite angles are congruent 4. m WW = m NN and m II = m SS 4. Definition of congruent angle 5. m NN = 90 o 5. Substitution (SN 2 and 4) 6. m WW + m II = 180 6. Consecutive angles are supplementary 7. 90 + m II = 180 7. Substitution (SN 2 and 6) 8. 90 = 90 8. Reflexive Property 9. m II = 90 9. Substitution (SN 7 and 8) 10. m SS = 90 10. Substitution (SN 4 and 9) 11. II, NN, and SS are right angles 11. If the measure of an angle is 90, then it is a right angle 12. WINS is a rectangle 12. Definition of rectangle Note: SN is for Statement Number 5

Theorem 2: The diagonals of a rectangle are congruent. Given: WINS is a rectangle with diagonals WWWW and SSSS. Prove: WWWW SSSS Proof: W S I N 1. WINS is a rectangle with diagonals WWWW and SSSS. 1. Given 2. WWWW IIII 2. Opposite sides of a parallelogram are congruent 3. WWWWWW and IIIIII are right angles 3. Theorem 1 4. WWWWWW and IIIIII 4. All right angles are congruent 5. SSSS NNNN 5. Reflexive Property 6. WSN INS 6. SAS Congruence Postulate 7. WWWW SSSS CPCTC (Corresponding Parts of Congruent Triangles are Congruent) Theorem 3: The diagonals of a rhombus are perpendicular. Given: Rhombus ROSE Prove: RRRR OOOO Proof: R E H O S 1. Rhombus ROSE 1. Given 2. OOOO RRRR 2. Definition of Rhombus 3. RRRR and EEEE bisect each other 3. The diagonals of a parallelogram bisect each other 4. H is the midpoint of RRRR 4. EEEE bisects RRRR at H 5. RRRR HHHH 5. Definition of midpoint 6. OOOO OOOO 6. Reflexive Property 7. RHO SHO 7. SSS Congruence Postulate 8. RRRRRR SSSSSS 8. CPCTC 9. RRRRRR and SSSSSS are right angles 9. RRRRRR and SSSSSS form a linear pair and are congruent 10. RRRR OOOO 10. Perpendicular lines meet to form right angles 6

Theorem 4: Each diagonal of a rhombus bisects opposite angles Given: Rhombus VWXY Prove: 1 2 3 4 Proof: 1. Rhombus VWXY 1. Given 2. YYYY VVVV ; WWWW XXXX 2. Definition of Rhombus 3. WWWW YYYY 3. Reflexive Property 4. YVW WXY 4. SSS Congruence Postulate 5. 1 2 ; 3 4 5. CPCTC What s More Activity 3: Answer the following statements with emoticons. Draw if it s always true, if it s sometimes true or if it s never true. 1. A square is a rectangle 2. A rhombus is a square 3. A parallelogram is a square 4. A rectangle is a rhombus 5. A parallelogram is a square 6. A parallelogram is a rectangle 7. A quadrilateral is a parallelogram 8. A square is a rectangle and a rhombus 9. An equilateral quadrilateral is a rhombus 10. An equiangular quadrilateral is a rectangle 7

Activity 3.1: Indicate with a check ( ) mark in the table below the property that corresponds to the given quadrilateral. Property 1. All sides are congruent 2. Opposite sides are parallel 3. Opposite sides are congruent 4. Opposite angles are congruent 5. Opposite angles are supplementary 6. Diagonals are congruent 7. Diagonals bisect each other 8. Diagonals bisect opposite angles 9. Diagonals are perpendicular to each other 10. A diagonal divides a quadrilateral into two congruent triangles Quadrilaterals Parallelogram Rectangle Rhombus Square What I Have Learned Activity 4: Show Me! Directions: Complete the table below by choosing the correct answer on the clouds. Given: AAAA DDDD, BBBBBB DDDDDD Prove: ABCD is a parallelogram B C A D AAAA DDDD, BBBBBB DDDDDD AAAA BBBB, BBBBBB DDDDDD ABCD is a parallelogram ABCD is a square SAS congruence postulate SSS congruence postulate Reflexive Property 8

1. 1. Given 2. AAAA AAAA 2. 3. AAAAAA CCCCCC 3. 4. BBBB AAAA 4. CPCTC 5. 5. Definition of a parallelogram What I Can Do Activity 5: Build! Build! Build! Prove! Prove! Prove! Directions: Read and answer the problem below. Four boy scouts are planning to build a rectangular tent. Scout Daryl said that the measurement from pole A to pole C is the same as the measurement from pole B to D. Another scout doesn t seem to agree to this, how will scout Daryl prove that pole A to C is congruent to pole B to D, given that the two parallel sides of the tent is also congruent? Given: Rectangle ABCD, AAAA BBBB Prove: AAAA BBBB A B D C Help scout Daryl prove that AAAA BBBB, by completing the table below. 1. 1. Given 2. DDDD CCCC 2. 3. AAAAAA and BBBBBB are right angles 3. Definition of a rectangle 4. AAAAAA BBBBBB 4. All right angles are congruent 5. AAAAAA BBBBBB 5. 6. AAAA BBBB 6. 9

Activity 5.1: Name It, to Win It! Directions: Name all the parallelograms that possess the given. Write Rh for rhombus, S for square, Rc for rectangle and ALL if it possesses the three. Statement 1. All sides are congruent 2. Diagonals bisect each other 3. Consecutive angles are congruent 4. Opposite angles are congruent 5. The diagonals are perpendicular and congruent (Rh, S, Rc or All) Assessment Directions: Read the questions carefully. Encircle the letter of the best answer. 1. In quadrilateral RSTW, diagonals RRRR and SSSS are perpendicular bisectors of each other. Quadrilateral RSTW must be a: I. Rectangle II. Rhombus III. Square a. I c. II and III b. II d. I, II and III 2. Which of the following is true about theorem on rectangle? a. The diagonals of a rectangle are parallel b. The diagonals of a rectangle are congruent c. If a parallelogram has a right angle, then it has five right angles d. If a parallelogram has one acute angle, then the parallelogram is a rectangle 3. Which of the following is NOT a way to prove a quadrilateral is a parallelogram? a. Show the diagonals of a rhombus are perpendicular b. Show both sets of opposite angles are congruent c. Show one set of opposite sides of the quadrilateral is both congruent and parallel d. Show one set of opposite sides is congruent 4. How many congruent triangles are formed when a diagonal of parallelogram is drawn? a. 1 b. 2 c. 3 d. 4 5. What does CPCTC in mathematics stands for? a. Corresponding parts of congruent triangles are congruent b. Corresponding parts of congruent triangles are coherent c. Corresponding parts of convex triangles are congruent d. Corresponding parts of convex triangles are coherent 10

6. Which is the correct proof on the given situation below: Given: Rhombus VENI Prove: 6 7 8 9 A. 1. Rhombus VENI 1. Given 2. IIII VVVV ; EEEE NNNN 2. Definition of Rhombus 3. IIII EEEE 3. Transitive Property 4. IVE ENI 4. SSS Congruence Postulate 5. 6 7 ; 8 9 5. CPCTC B. 1. Rhombus VENI 1. Given 2. VVVV IIII ; EEEE EEEE 2. Definition of Rhombus 3. IIII EEEE 3. Reflexive Property 4. IVN ENV 4. SSS Congruence Postulate 5. 6 7 ; 8 9 5. CPCTC C. 1. Rhombus VENI 1. Given 2. VVVV IIII ; EEEE EEEE 2. Definition of Parallelogram 3. IIII EEEE 3. Reflexive Property 4. IVN ENV 4. SAS Congruence Postulate 5. 6 7 ; 8 9 5. CPCTC D. 1. Rhombus VENI 1. Given 2. IIII VVVV ; EEEE NNNN 2. Definition of Rhombus 3. IIII EEEE 3. Reflexive Property 4. IVE ENI 4. SSS Congruence Postulate 5. 6 7 ; 8 9 5. CPCTC 11

For numbers 7 and 8: Complete the two-column proof table below. H T A E Given: HATE is a rectangle, with diagonals HHHH and TTTT Prove: HHHH TTTT Proof: STATEMENTS REASONS 1. HATE is a rectangle, with diagonals HHHH and TTTT 1. Given 2. HHHH AAAA 2. Opposite sides of a parallelogram are congruent 3. HHHHHH and AAAAAA are right angles 3. Theorem no. 1 4. HHHHHH and AAAAAA 4. All right angles are congruent 5. TTTT EEEE 5. (8) 6. (7) 6. SAS congruence postulate 7. HHHH TTTT 7. CPCTC 7. 8. a. HTE AET b. TAH ATE c. AHT ATH d. HTE EHA a. Transitive property b. Reflexive property c. SSS congruence postulate d. SAS congruence postulate For numbers 9 and 10: Directions: Complete the two-column proof with the following given below. Show Me! G A Given: Rhombus GABE Y Prove: GGGG AAAA E B 12

1. 1. Given 2. AAAA GGGG 2. 3. 3. The diagonals of a parallelogram bisect each other 4. Y is the midpoint of GGGG 4. 5. GGGG YYYY 5. Definition of midpoint 6. AAAA YYYY 6. 7. 7. SSS Congruence Postulate 8. GGGGGG BBBBBB 8. CPCTC 9. GGGGGG and BBBBBB are right angles 9. GGGGGG and BBBBBB form a linear pair and are congruent 10. GGGG AAAA 10. Perpendicular lines meet to form right angles 9. Which is best for statements 1, 3 and 7? a. 1. Rhombus GABE, 3. GGGG and YYYY bisect each other, 7. GYE BYA b. 1. Rhombus GABE, 3. GGGG and EEEE bisect each other, 7. GYE BYA c. 1. Rhombus GABE, 3. GGGG and EEEE bisect each other, 7. GYA BYA d. 1. Rhombus GABE, 3. GGGG and EEEE bisect each other, 7. GYA BYA 10. Which is best for statements 2, 4 and 6? a. 2. Definition of Rhombus, 4. EEEE bisects YYYY at Y, 6. Reflexive Property b. 2. Definition of Rhombus, 4. EEEE bisects GGGG at Y, 6. Reflexive Property c. 2. Definition of Rectangle, 4. EEEE bisects YYYY at Y, 6. Reflexive Property d. 2. Definition of Rectangle, 4. EEEE bisects GGGG at Y, 6. Reflexive Property 13

ADDITIONAL ACTIVITY Activity 8: Especially for You Directions: Do the procedures below and answer the questions that follow. Materials: bond paper, pencil, ruler, protractor, and compass Procedure: 1. Draw square GOLD. (Note: draw a square based on its definition: parallelogram with 4 congruent sides and 4 right angles.) 2. Draw diagonals GGGG and OOOO that meet at C. 3. Use a ruler to measure the segments indicated in the table. 4. Use a protractor to measure the angles indicated in the table What to measure Measurement GGGGGG GGGG and OOOO GGGGGG and OOOOOO GGGGGG and OOOOOO GGGGGG and LLLLLL Questions: 1. What is the measure of GGGGGG? If GGGGGG is a right angle, can you consider square a rectangle? If yes, what theorem on rectangle justifies that a square is a rectangle? 2. What can you say about the length of GGGG and OOOO? If GGGG and DDDD have the same measures, can you consider a square a rectangle? If yes, what theorem on rectangles justifies that a square is a rectangle? 3. What can you say about the measures of GGGGGG and OOOOOO? If GGGG and DDDD meet to form right angles, can you consider a square a rhombus? If yes, what theorem on rhombus justifies that a square is a rhombus? 4. What can you say about the measures of GGGGGG and OOOOOO as a pair and GGGGGG and LLLLLL as another pair? If GGGG divides opposite angles equally, can you consider a square a rhombus? If yes, what theorem on rhombuses justifies that a square is a rhombus? 14

References Bryant, Merden L., Leonides E. Bulalayao, Melvin M. Callanta, Jerry D. Cruz, et al. Mathematics Learner s Material 9. Pasig City: Department of Education, 2014. Bryant, Merden L., Leonides E. Bulalayao, Melvin M. Callanta, Jerry D. Cruz, et al. Mathematics Teachers Guide 9. Pasig City: Department of Education, 2014. The Organic Chemistry Tutor. Proving Parallelograms with Two Column Proofs Geometry. December 26, 2017. www.youtube.com/watch?v=mqjnpfnb2og. MrPilarski. Special Parallelograms Part 2 Rhombus and Rectangle Proofs Geometry Help. January 25, 2012. www.youtube.com/watch?v=-zkcfd64hom. Development Team Writer: Veni Lester B. Manlapaz Zamboanga Sibugay National High School Editor/QA: Eugenio E. Balasabas Ressme M. Bulay-og Mary Jane I. Yeban Reviewer: Gina I. Lihao EPS-Mathematics Illustrator: Layout Artist: Management Team: Evelyn F. Importante OIC-CID Chief EPS Jerry c. Bokingkito OIC-Assistant SDS Aurelio A. Santisas, CESE OIC- Assistant SDS Jenelyn A. Aleman, CESO VI OIC- Schools Division Superintendent 15