MATH 3 part 2

9. Proof: Reason Statement 1. Given 2. Vertical ∠ s are ≅ 1. LO ≅ MO 3. LA congruence 2. ∠ LOJ ≅ ∠ MOK 4. CPCTC 3. ∆ LOJ ≅ ∆ MOK 4. ∠ J ≅ ∠ K Reason10. Proof: 1. Given Statement 2. Reflexivity 3. SAS congruence 1. PQ ≅ RS 4. CPCTC ∠ QPR ≅ ∠ SRP 2. PR ≅ PR 3. ∆ QPR ≅ ∆ SRP 4. SP ≅ QR 24

Module 3Geometric Relations What this module is about This module is about the angles formed by parallel lines (//) cut by atransversal. You will learn to determine the relation between pairs of anglesformed by parallel lines cut by a transversal and solve problems involvingsegments and angles. What you are expected to learnThis module is designed for you to:1. identify the angles formed by parallel lines cut by a transversal.2. determine the relationship between pairs of angles formed by parallel lines cut by a transversal: • alternate interior angles • alternate exterior angles • corresponding angles • angles on the same side of the transversal3. solve problems using the definition and properties involving relationships between segments and between angles.How much do you knowThe figure below shows lines m // n with t as transversal.Name: 7 m 86 n 1. 4 pairs of corresponding angles 5 2. 2 pairs of alternate interior angles 3. 2 pairs of alternate exterior angles 3 42 1 t

4. 2 pairs of interior angles on the same side of the transversal 5. 2 pairs of exterior angles on the same side of the transversalUsing the same figure:6. Name all numbered angles congruent to ∠ 7.7. Name all numbered angles congruent to ∠ 4.8. Name all numbered angles supplementary to ∠ 8, to ∠ 7.9. Name all numbered angles supplementary to ∠ 3, to ∠ 4.10. Name the pairs of equal angles and supplementary angles in the figure.Given: AB // CD B AD // BC A C DIn the figure below, AAB ⊥ BD, B 1 CDF ⊥ BD, 2BC // DE E 3D 4 F11. Is m ∠ 2 = m ∠ 3? Why?12. ∠ 2 is a complement of _____ and ∠ 3 is a complement of ____.13. Is m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4? Why? 2

Find the value of x in each of the following figure: 14. 70o xo 55o15. xo 140o 110o16. 3x xo17. 4xxo 3

In the figure, write down the pairs of parallel lines and the pairs ofcongruent angles.18. CB A D F E G19. If m ∠ 3 = 135, find the measure of each angle in the figure. 12 m 43 56 n 87 t20. If m ∠ 6 = 85, find the measure of each numbered angle in the figure, a // b and c // d.67 9 10 m58 12 11 43 13 14 n 12 16 15 cd 4

What you will do Lesson 1Angles Formed by Parallel Lines Cut by a Transversal In the rectangular solid below, AB and CD are coplanar in plane x. ABand EF are coplanar in plane y. EF and HF are coplanar in plane z.C y G B F x zD H A E Line E intersect AB and CD at two different points. Line E is a transversalof lines AB and CD.Definitions: Coplanar lines are lines that lie in one plane. Parallel lines are two lines that are coplanar and do not intersect. Transversal is a line that intersects two or more lines at different points. EB A CD Line E intersect AB and CD at 2 different points. Line E is a transversal oflines AB and CD. 5

Examples:1. In the figure, lines AB and CD 12 are parallel lines cut by transversal A 43 B line t. The angles formed are:Angles 1, 2, 7 and 8 are exterior angles C 56 DAngles 3, 4, 5, and 6 are interior angles 87The pairs of corresponding angles are: t ∠ 1 and ∠ 5 ∠ 2 and ∠ 6 ∠ 4 and ∠ 8 ∠ 3 and ∠ 7The pairs of alternate interior angles are: ∠ 3 and ∠ 5 ∠ 4 and ∠ 6The pairs of alternate exterior angles are: ∠ 1 and ∠ 7 ∠ 2 and ∠ 8The pairs of exterior angles on the same side of a transversal (SST) are: ∠ 1 and ∠ 8 ∠ 2 and ∠ 7The pairs of interior angles on the same side of a transversal (SST) are: ∠ 4 and ∠ 5 ∠ 3 and ∠ 62. Given: m // n, s is the transversal. The pairs of angles formed are:a. Corresponding angles: 34 m 65 n ∠ 3 and ∠ 9 ∠ 6 and ∠ 12 9 10 ∠ 4 and ∠ 10 14 11 ∠ 5 and ∠ 11 s 6

b. Alternate exterior angles ∠ 3 and ∠ 11 ∠ 4 and ∠ 12c. Alternate interior angles ∠ 6 and ∠ 10 ∠ 5 and ∠ 9d. Interior angles on the same side of the transversal (SST). ∠ 6 and ∠ 9 ∠ 5 and ∠ 10e. Exterior angles on the same side of the transversal (SST). ∠ 3 and ∠ 12 ∠ 4 and ∠ 113. m // n, t is the transversal. 12 m 87In the figure, name and identify he pairs of angles formed: 9 10 n 12 11a. Corresponding angles: t ∠ 1 and ∠ 9 ∠ 8 and ∠ 12 ∠ 2 and ∠ 10 ∠ 7 and ∠ 11b. Alternate interior angles ∠ 8 and ∠ 10 ∠ 7 and ∠ 9 7

c. Alternate exterior angles ∠ 1 and ∠ 11 ∠ 2 and ∠ 12 d. Exterior angles on the same side of the transversal (SST). ∠ 1 and ∠ 12 ∠ 2 and ∠ 11 e. Interior angles on the same side of the transversal (SST). ∠ 8 and ∠ 9 ∠ 7 and ∠ 10Try this out 1. In the figure, lines g // h and is cut by line k. 41 g 32 85 h76 kName and identify the pairs of angles formed2. In the figure, q // r and s // t.15 16 78 q14 13 6511 12 34 r10 9 21 stName and identify the pairs of angles formed. 8

3. In the figure, u // w. Lines x and y are transversals.65 12 u7 8 43 9 w 10 14 x 11 13 12 yName and identify the pairs of angles formed4. In the figure, identify and name the pairs of parallel lines and its transversal. AB CD G EF5. In the figure, identify and name the pairs of parallel lines and its transversal / transversals. ab c d e 9

Lesson 2Relationship Between Pairs of Angles Formed by Parallel Lines Cut by a Transversal If two lines are cut by a transversal, then: a. corresponding angles are congruent b. alternate interior angles are congruent c. alternate exterior angles are congruent d. interior angles on the same side of a transversal are supplementary e. exterior angles on the same side of a transversal are supplementaryExamples: 1. Given: p // q, r is a transversal Figure: 14 p 23 58 q 67 rWhat is the measure of each numbered angles if m ∠ 1 = 120? Give thereason for your answer.Answers:If ∠ 1 = 120o, ∠ 5 = 120o Corresponding angles are ≅ .If ∠ 5 = 120o, ∠ 3 = 120o Alternate interior angles are ≅If ∠ 3 = 120o, ∠ 7 = 120o Corresponding angles are ≅If ∠ 7 = 120o, ∠ 4 = 60o Exterior angles on the same side of a transversal are supplementaryIf ∠ 4 = 60o, ∠ 8 = 60o Corresponding angles are ≅If ∠ 8 = 80o, ∠ 2 = 60o Alternate interior angles are ≅If ∠ 2 = 60o, ∠ 6 = 60o Corresponding angles are ≅ 10

2. Given: In the figure, if m ∠ 1 = 105, determine the measures of the other numbered angles. Justify your answers.Figure: t 12 A 43 B 56 D C 87 Corresponding angles are ≅ .Answers: Alternate interior angles are ≅ Corresponding angles are ≅ If ∠ 1 = 105o, ∠ 5 = 105o Exterior angles on the same side If ∠ 5 = 105o, ∠ 3 = 105o of a transversal are supplementary If ∠ 3 = 105o, ∠ 7 = 105o Alternate exterior angles are ≅ If ∠ 7 = 105o, ∠ 2 = 75o Corresponding angles are ≅ Alternate interior angles are ≅ If ∠ 2 = 75o, ∠ 8 = 75o If ∠ 8 = 75o, ∠ 4 = 75o If ∠ 4 = 75o, ∠ 6 = 75o3. Given: m // n, s and t are the transversals tsIf m ∠ 5 = 110 and m ∠ 12 = 90, determine 12 3 4 mthe measures of the other numbered ∠ s. 87 6 5Justify your answer. 10 9 11 n 14 12 13Answers:If m ∠ 5 = 110, m ∠ 12 + m ∠ 13 = 110Since m ∠ 12 = 90, m ∠ 13 = 20. Corresponding angles are ≅m ∠ 4 = 70 ∠ 4 and ∠ 5 are supplementaryIf m ∠ 4 = 70, m ∠ 11 = 70 Corresponding angles are ≅m ∠ 3 = 110 ∠ 3 and ∠ 5 are vertical angles 11

If m ∠ 3 = 110, m ∠ 9 + m ∠ 10 = 110 Corresponding angles are ≅If m ∠ 9 = 90, m ∠ 7 = 90 Alternate interior angles are ≅If m ∠ 12 = 90, m ∠ 2 = 90 Exterior angles on the same side of a transversal are supplementarySince m ∠ 9 = 90, m ∠ 1 = 90 Corresponding angles are ≅If m ∠ 13 + m ∠ 14 = m ∠ 8, m ∠ 8 = 90 Corresponding angles are ≅Therefore, the measures of the numbered angles are:∠ 1 = 90o ∠ 8 = 90o∠ 2 = 90o ∠ 9 = 90o∠ 3 = 110o ∠ 10 = 20o∠ 4= 70o ∠ 11 = 70o∠ 5 = 110o ∠ 12 = 90o∠ 6 = 70o ∠ 13 = 20o∠ 7 = 90o ∠ 14 = 70oTry this out1. If m ∠ 6 = 85, find the measure of the other numbered angles. Justify your answers.Given: a // b, c is the transversalFigure: ab 12 56 c 43 87 12

2. If m ∠ 10 = 118 and m ∠ 4 = 85, find the measures of the other numbered angles. Justify your answers.Given: f // g, r // q 87 56 9 10 f 12 11 43 13 14 g 12 16 15 rq3. In the figure, u // w. Lines x and y are transversals. If m ∠ 8 = 62, m ∠ 14 = 90, find the measures of the other numbered ∠ s. Justify your answers.Figure: 65 12 u 7 8 43 9 w 10 14 x 11 13 12 yForm an equation in x and solve the equation.4. 2xo xo 13

5. xo 120o 100o Let’ summarizeThe angles formed by parallel lines cut by a transversal are:1. corresponding angles2. alternate interior angles3. alternate exterior angles4. exterior angles on the same side of a transversal5. interior angles on the same side of a transversalThe relationship of the angles formed by parallel lines cut by a transversalare:1. pairs of corresponding s are ≅2. pairs of alternate interior angles are ≅3. pairs of alternate exterior angles are ≅4. pairs of interior angles on the same side of a transversal are supplementary5. pairs of exterior angles on the same side of a transversal are supplementary 14

What have you learnedThe figure below shows lines m // n with t as transversal.Figure: t 12 m 43 56 n 87Name:1. 4 pairs of corresponding angles.2. 2 pairs of alternate interior angles.3. 2 pairs of alternate exterior angles.4. 2 pairs of interior angles on the same side of a transversal.5. 2 pairs of exterior angles on the same side of a transversal. Using the same figure:6. Name all numbered angles congruent to ∠ 7.7. Name all numbered angles congruent to ∠ 4.8. Name all numbered angles supplementary to ∠ 8, ∠ 7.9. Name all numbered angles supplementary to ∠ 3, ∠ 4.10. Name the pairs of equal angles and supplementary angles in the figure. Given: AB // CD Figure: AD // BC A B DC 15

11. – 13. In the figure, AB ⊥ BD, DF ⊥ BD, BC // DEFigure: A C D 2 1 3 B 4 E F11. Is m ∠ 2 = m ∠ 3? Why?12. ∠ 2 is a complement of _____ and ∠ 3 is a complement of _____.13. Is m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4? Why?Find the value of x in each of the following figures.14. 50o xo xo15. 100o 120o xo 16

16. 5x xo17. xo 4x In the figure, Write down the pairs of parallel lines and the pairs of congruent angles.18. B AFD C E 17

19. If m ∠ 1 = 135, find the measure of each angle in the figure below: t 12 m 43 56 n 8720. If m ∠ 6 = 75, find the measure of each numbered angle in the figure, a // b and c // d.67 9 10 a58 12 1143 13 14 b 12 16 15 c d 18

Answer keyHow much do you know1. ∠ 8 and ∠ 4, ∠ 7 and ∠ 3∠ 5 and ∠ 1, ∠ 6 and ∠ 22. ∠ 5 and ∠ 3, ∠ 4 and ∠ 63. ∠ 8 and ∠ 2, ∠ 7 and ∠ 14. ∠ 5 and ∠ 4, ∠ 6 and ∠ 35. ∠ 8 and ∠ 1, ∠ 7 and ∠ 26. ∠ 3, ∠ 17. ∠ 8, ∠ 68. To ∠ 8: ∠ 1, ∠ 5, ∠ 7; To ∠ 7: ∠ 2, ∠ 8, ∠ 69. To ∠ 3: ∠ 6, ∠ 4, ∠ 2; To ∠ 4: ∠ 5, ∠ 3, ∠ 110. ∠ A ≅ ∠ C, ∠ B ≅ ∠ D, ∠ A supplement ∠ B, ∠ B supplement ∠ C ∠ C supplement ∠ D, ∠ D supplement ∠ A11. They are alternate interior ∠ s12. ∠ 1, ∠ 413. Yes, because they are right angles.14. x = 55o15. x = 110o16. x = 45o17. x = 36o18. AB // DE; CD // FG; ∠ ACD ≅ ∠ EDC; ∠ EFG ≅ ∠ CDF19. ∠ 2 = 45o; ∠ 4 = 45o; ∠ 1 = 135o; ∠ 7 = 135o; ∠ 8 = 45o; ∠ 6 = 45o; ∠ 5 = 135o20. ∠ 4 = 85o; ∠ 5 = 95o; ∠ 1 = 95o; ∠ 7 = 95o; ∠ 3 = 95o; ∠ 8 = 85o; ∠ 2 = 85o ∠ 10 = 95o; ∠ 9 = 85o; ∠ 12 = 95o; ∠ 11 = 85o; ∠ 13 = 85o; ∠ 14 = 95o; ∠ 16 = 95o; ∠ 15 = 85o 19

Try this outLesson 11. ∠ 4 and ∠ 8; ∠ 3 and ∠ 7; ∠ 1 and ∠ 5; ∠ 2 and ∠ 6 are corresponding angles; ∠ 4 and ∠ 7; ∠ 1 and ∠ 6 are exterior angles on SST; ∠ 3 and ∠ 5; ∠ 2 and ∠ 8 are alternate interior angles; ∠ 4 and ∠ 7; ∠ 1 and ∠ 6 are exterior angles on SST; ∠ 3 and ∠ 5; ∠ 2 and ∠ 8 are alternate interior angles; ∠ 3 and ∠ 8; ∠ 2 and ∠ 5 are interior angles on SST; ∠ 4 and ∠ 6; ∠ 1 and ∠ 8 are exterior angles on SST;2. ∠ 1 and ∠ 5; ∠ 4 and ∠ 8; ∠ 2 and ∠ 6; ∠ 3 and ∠ 7 are corresponding angles; ∠ 4 and ∠ 6; ∠ 3 and ∠ 5 are alternate interior angles; ∠ 7and ∠ 2; ∠ 8 and ∠ 1 are exterior angles on SST; ∠ 6 and ∠ 3; ∠ 5 and ∠ 4 are interior angles on SST; ∠ 5 and ∠ 13; ∠ 6 and ∠ 14; ∠ 7 and ∠ 15; ∠ 8 and ∠ 16 are corresponding angles; ∠ 8 and ∠ 15; ∠ 5 and ∠ 14 are exterior angles on SST; ∠ 7and ∠ 16; ∠ 6 and ∠ 13 are interior angles on SST; ∠ 16 and ∠ 6; ∠ 13 and ∠ 7 are alternate interior angles; ∠ 15 and ∠ 5; ∠ 14 and ∠ 8 are alternate exterior angles; ∠ 15 and ∠ 11; ∠ 14 and ∠ 10; ∠ 16 and ∠ 12; ∠ 13 and ∠ 9 are corresponding angles; ∠ 14 and ∠ 11; ∠ 13 and ∠ 12 are interior angles on SST; ∠ 15and ∠ 10; ∠ 16 and ∠ 9 are exterior angles on SST; ∠ 4 and ∠ 12; ∠ 13 and ∠ 11 are alternate interior angles; ∠ 15 and ∠ 9; ∠ 16 and ∠ 10 are alternate exterior angles; ∠ 3 and ∠ 11; ∠ 4 and ∠ 12; ∠ 2 and ∠ 10; ∠ 1 and ∠ 9 are corresponding angles; ∠ 12 and ∠ 2; ∠ 3 and ∠ 9 are alternate interior angles; ∠ 11 and ∠ 1; ∠ 4 and ∠ 10 are alternate exterior angles; 20

∠ 1and ∠ 10; ∠ 4 and ∠ 11 are exterior angles on SST; ∠ 3 and ∠ 12; ∠ 2 and ∠ 9 are interior angles on SST;3. ∠ 1 and ∠ 9 + ∠ 10; ∠ 4 and ∠ 11 are corresponding angles; ∠ 2 and ∠ 14; ∠ 3 and ∠ 12 + ∠ 13 are corresponding angles; ∠ 4 and ∠ 14; ∠ 3 and ∠ 9 + ∠ 10 are alternate interior angles; ∠ 1 and ∠ 12 + ∠ 13; ∠ 2 and ∠ 11 are alternate exterior angles; ∠ 1 and ∠ 11; ∠ 2 and ∠ 12 + ∠ 13 are exterior angles on SST; ∠ 4 and ∠ 9 + ∠ 10; ∠ 3 and ∠ 14 are interior angles on SST; ∠ 6 and ∠ 10; ∠ 7 and ∠ 11; ∠ 5 and ∠ 9 + ∠ 14; ∠ 8 and ∠ 13 are corresponding angles; ∠ 7 and ∠ 9 + ∠ 14; ∠ 8 and ∠ 10 are alternate interior angles; ∠ 6 and ∠ 13; ∠ 5 and ∠ 11 + ∠ 12 are alternate exterior angles; ∠ 6and ∠ 11 + ∠ 12; ∠ 5 and ∠ 13 are exterior angles on SST; ∠ 7 and ∠ 10; ∠ 8 and ∠ 9 + ∠ 14 are interior angles on SST;4. AB // CD with BC as transversal CD // FE with DE as transversal DE // FG with FE as transversal5. a // b, e, c and d are the transversals6. c // d, a, b and c are the transversalsLesson 21. ∠ 2 = 85o; ∠ 5 = 95o; ∠ 1 = 95o; ∠ 7 = 95o; ∠ 3 = 95o; ∠ 8 = 85o; ∠ 4 = 85o2. ∠ 12 = 118o; ∠ 9 = 62o; ∠ 11 = 62o; ∠ 16 = 118o; ∠ 13 = 62o; ∠ 14 = 118o; ∠ 15 = 62o ∠ 8 = 85o; ∠ 6 = 85o; ∠ 2 = 85o; ∠ 1 = 95o; ∠ 5 = 95o; ∠ 3 = 95o; ∠ 1 = 95o3. ∠ 8 = 62o; ∠ 6 = 62o; ∠ 10 = 62o; ∠ 13 = 118o; ∠ 9 = 28o; ∠ 12 = 28o; ∠ 11 = 90o; ∠ 13 = 62o; ∠ 7 = 118o; ∠ 5 = 118o; ∠ 1 = 90o; ∠ 2 = 90o ; ∠ 3 = 90o; ∠ 4 = 90o 21

4. x = 60o5. x = 140oWhat have you learned1. ∠ 1 and ∠ 5, ∠ 4 and ∠ 8, ∠ 2 and ∠ 6, ∠ 3 and ∠ 72. ∠ 4 and ∠ 6, ∠ 3 and ∠ 53. ∠ 1 and ∠ 7, ∠ 2 and ∠ 84. ∠ 4 and ∠ 5, ∠ 3 and ∠ 65. ∠ 1 and ∠ 8, ∠ 2 and ∠ 76. ∠ 3, ∠ 5, ∠ 17. ∠ 8, ∠ 6, ∠ 28. ∠ 8: ∠ 5, ∠ 7, ∠ 1; ∠ 7: ∠ 2, ∠ 6, ∠ 89. ∠ 3: ∠ 6, ∠ 4, ∠ 2; ∠ 4: ∠ 5, ∠ 3, ∠ 110. ∠ A ≅ ∠ C; ∠ B ≅ ∠ D; ∠ A supplementary ∠ B ∠ B supplementary ∠ C ∠ C supplementary ∠ D ∠ A supplementary ∠ D11. Yes, they are alternate interior angles12. ∠ 1, ∠ 413. Yes, because their sum is equal to 90o14. x = 65o15. x = 140o16. x = 30o17. x = 36o18. BA // CF, BC // DE, ∠ ABC ≅ ∠ BCF; ∠ BCD ≅ ∠ CDE19. ∠ 2 = 45o, ∠ 3 = 135o, ∠ 4 = 45o, ∠ 5 = 135o, ∠ 6 = 45o, ∠ 7 = 135o, ∠ 8= 45o20. ∠ 4 = 75o, ∠ 1 = 105o, ∠ 5 = 105o, ∠ 7 = 105o, ∠ 3 = 105o, ∠ 8 = 75o,∠ 2 = 75o, ∠ 9 = 75o, ∠ 10 = 105o, ∠ 12 = 105o, ∠ 11 = 75o 22

Module 3 QuadrilateralsWhat this module is aboutThis module is about Quadrilaterals. As you go over the exercises,you will develop skills in identifying Quadrilaterals and their parts andability to appreciate their application in daily life. Treat the lessonwith fun and take time to go back if you think you are at a loss.What you are expected to learn E This module is designed for you to learn 1. illustrate a Quadrilateral and its parts 2. illustrate the different kinds of QuadrilateralsHow much do you know?Write the letter of the correct answer Z YN O1. Quadrilateral ZENY is a ____. a. parallelogram b. trapezoid c. trapezium d. rectangle2. A diagonal of quadrilateral ZENY is ____.a. ZE b. ZN c. ZO d. ZY d. ∠O3. ∠Y is opposite angle ____.a. ∠Z b. ∠E c. ∠N4. DH & ET are the ____ of the Quadrilateral DETH a. median b. altitude c. bases d. legs5. If BH ≅ ET then BETH is a/an ____.a. rectangle b. trapeziumc. isosceles trapezoid d. square6. BH ≅ ET are the _____ of Quadrilateral BETH.a. legs b. bases c. median d. altitude

7. whish is the median of quadrilateral DETHa. BH b. MP c. EA d. DH8. What kind of parallelogram is quadrilateral DETH?a. rhombus b. square c. parallelogram d. rectangle9. Using the figure at the right which is a rhombus?a. ACGH b. CEFGc. BDGH d. AEFH A B CD E H G F10. Which is a rhombus and a rectangle? d. BDGH a. AEFH b. CEFG c. ACGHWhat you will do Lesson 1 Identifying and Naming Quadrilaterals.A quadrilateral is a polygon of four sidesExample: M O D E O O LA DE RP M EYou ca name a quadrilateral by its vertices. The order of vertives is veryimportant. You read or write the four letters clockwise or counterclockwise.Examples: ROYou can name a quadrilateral at theRight as.ROSE or OSER or SERO or EROSOr RESO or ESOR or SORE ES

The name of this quadrilateral can be: K L MKLMN or LMNK or MNKL or NKLMOr NMLK or MLKN or LKNM or KNML NTry this out. A. Which of the following is a quadrilateral or not.1) 3) 5)2) 4) 6)7) 9)8) 10) B. Name the following quadrilaterals.C1) O 2) H O DL E P3)B E 4) O PHA TS5) A 6) B M LR A YY

7)M A 8)A BY ND C9) O R 10) S PR QSE C. Name & Identify the Quadrilaterals in the figure.AB C D EEE hhhdhjhsjsjj Name 10 Quadrilaterals.

Lesson 2 Parts of a QuadrilateralA quadrilateral has the following parts:4 sides4 vertices4 angles2 diagonalsYou can take a look on quadrilateral LOVE L O V• the sides are: LO , OV , EV , LE• the vertices are: L, O, V, E• the angles are ∠L , ∠O , ∠V , ∠E• the diagonals are LV & OE Ediagonals are segments joining opposite vertices. The vertices E and O; L and V are opposite vertices. Vertices L and O, O andV, V and E, E and L are consecutive vertices. Two sides with a common vertex like LO and OV are consecutive sides.So, OV and VE , VE and EL , EL and LO are other pairs of consecutive sides.On the otherhand, LO and EV , OV and LE are opposite sides. Two angles with a common side like ∠L and ∠O are consecutive angles,the others are ∠O and ∠V , ∠V and ∠E , ∠E and ∠L , on the other hand, ∠Eand ∠O ; ∠L and ∠V are opposite angles.

Try this out C OA. Using quadrilateral COLA, identify L O 1. two pairs of opposite vertices A P 2. two pairs of opposite angles M P Q 3. two pairs of opposite sides E 4. one pair of diagonals 5. four pairs of consecutive vertices R 6. four pairs of consecutive angles 7. four pairs of consecutive sides 8. four sides of quadrilateral POEM 9. four angles of quadrilateral POEM 10. two diagonals that can be drawn in quadrilateral POEMB. Fill the blanks:1) ______ the vertex opposite S2) ______ is the opposite side of PS S3) ∠Q is opposite _________4) PQ and QR are _______ sides.5) The diagonals that can be drawn are QS and ______ B C AD 6) BC and CD are ___________ sides. 7) B and C are consecutive vertices, B and _____ are also consecutive vertices. 8) AC and _____ are the diagonals of quadrilateral ABCD. 9) ∠A and ∠D are ______ angles.10) ∠A and _____ are opposite angles.

C. Choose the letter of the correct answer. Use quadrilateral D E T H1. How many diagonals has quadrilateral DETH? D E a. one b. twoc. three d. four I2. The sides of quadrilateral DETH areDH , HT , TE and ______ Oa. HE b. DT c. DE d. DO3. ∠LH is opposite of what angle?a. ∠E b. ∠D c. ∠T d. ∠O H4. The opposite side of ET is _________a. DO b. DE c. HT d. DH5. The diagonals of quadrilateral DETH are _________a. DO & OT b. HE & DT c. DH & ET d. DE &HT Q R SGiven: quadrilateral PQRS6. How many vertices are there in Quadrilateral PQRS? a. one b. twoc. three d. four7. A pair of consecutive sides is PQ and ______a. QR b. RS c. PR d.P QS8. A pair of opposite vertices is P and ______a. R b. Q c. S d. M9. How many pairs of opposite ∠s has quadrilateral PQRS?a. one b. twoc. three d. four10. How many pairs of consecutive sides has quadrilateral PQRS?a. one b. twoc. three d. four

Lesson 3Parallels & PerpendicularsKinds of Quadrilaterals QuadrilateralTrapezium Trapezoid Parallelogram As you can see the diagram of the different kinds of quadrilaterals, youcan notice the characteristics of the sides of each quadrilateral. Before you proceed to the definition of each quadrilateral, you must knowfirst the meaning of the following:i. parallel lines two lines are parallel if they are coplanar and they doii. perpendicular lines not meet. two lines are perpendicular if they intersect and form a right angle.

Examples: B A two lines are parallel if they are coplanar1. D and they do not meet. C2. l1 is parallel to l2. In symbol l1 // l23. L2 L1 X R If XY intersects RS at O and ∠XOS is OS a right angle, then XY is perpendicular to RS , in symbol: XY ⊥ RS Y If ∠MAN is a right angle, then AM ⊥ AN4, AN

Try this out.A. Which of the following seem to be parallel? Write yes if it is and no if not.1) 7)2) 8)3) 9)4) 10)5)6)

B. Write parallel or perpendicular. 4) 1) 5) 2) 3) 6) A pair of lines of your pad paper 7) The corner of the blackboard 8) Railroad tracks 9) The grills 10) A pair of consecutive sides of a picture frameC. Are we parallel or perpendicular? 1) Two lines on a plane which do not meet. 2) Two intersecting lines which form a right angle. 3)4) AB & BD K L C5) FG & EH FI D6) KI & FG AE JG H B

7) PT & TU PV Q8) QR & PS TU R9) PV & TU O S10) BY & BO B D Y

Lesson 4 R Kinds of Quadrilaterals E D1. Trapezium M HOIf a quadrilateral has no parallel sides, then it is a trapezium. MORE is atrapezium.2. Trapezoid E P STIf a quadrilateral has exactly a pair of parallel sides, then it is a trapezoid.If HO // EP , then HOPE is a trapezoid.3. Parallelogram P OIf a quadrilateral has two pairs of parallel sides, then it is a parallelogram.If ST // PO and SP // PO then STOP is a parallelogram. W HA. Identify:______ 1. A quadrilateral whose opposite sides are parallel.______ 2. A quadrilateral with no parallel sides.______ 3. A quadrilateral with a pair of parallel sides.

______ 4. Quadrilateral WHEN. CX O______ 5. Quadrilateral COLD. X X D L______ 6. Quadrilateral SORE______ 7. Quadrilateral PEAS X______ 8. Quadrilateral TOME SXO E XR PE SA TX O X X EX M PE______ 9. Quadrilateral PEAK A E LN F

______ 10. Quadrilateral LOAF J K B. SelecAt the correct word from the set (trapezium, trapezoid, L parallelogram) N O R Q M RCD E ER P X W SU V H TY HG F SA TX O EX M K 1. quadrilateral DETH 2. quadrilateral DANS 3. quadrilateral SNLU 4. quadrilateral LOVU 5. quadrilateral JMRK 6. quadrilateral OMPV 7. quadrilateral RWCR 8. quadrilateral WGFC 9. quadrilateral PWGY 10. quadrilateral TERY C. Draw the following figures: O A 1. trapezium ZENY 2. trapezoid BETH 3. parallelogram LOVE 4. parallelogram with diagonals DL & BW 5. trapezoid with diagonals IE & MC

AF B D CE H GJ H M N KLTwo tents are fixed above. Give me: 6. 7. Tthwreoetrpaapreazlloeildosgrams 8. 9. 10.

Module 3 Geometry of Shape and Size What this module is about This module is about Quadrilaterals. As you go over theexercises, you will develop skills in identifying Quadrilaterals and theirparts. You will learn to appreciate the lesson as they are applied inyour daily life. What you are expected to learn This module is designed for you to learn to: 1. identify a Quadrilateral and name its parts 2. illustrate the different kinds of Quadrilaterals How much do you knowWrite the letter of the correct answer ZEYN1. Quadrilateral ZENY is a ____. a. parallelogram b. trapezoid c. trapezium d. rectangle2. A diagonal of quadrilateral ZENY is ____.a. ZE b. ZN c. ZO d. ZY3. ∠Y is opposite angle ____.a. ∠Z b. ∠E c. ∠N d. ∠O

Using the figure below: E DBMPHT4. DH & ET are the ____ of the Quadrilateral DETH a. median b. altitude c. bases d. legs5. If BH ≅ ET then BETH is a/an ____.a. rectangle b. trapeziumc. isosceles trapezoid d. square6. BH ≅ ET are the _____ of Quadrilateral BETH.a. legs b. bases c. median d. altitude7. Which is the median of quadrilateral DETH?a. BH b. MP c. EA d. DH8. What kind of parallelogram is quadrilateral DETH? a. rhombus b. square c. parallelogram d. trapezoid9. Using the figure below, which is a rhombus? a. ACGH b. CEFG c. BDGH d. AEFH A B CD E H GF10. Which is a rhombus and a rectangle? a. AEFH b. CEFG c. ACGH d. BDGH 2

What you will do Lesson 1 Identifying and Naming Quadrilaterals.A quadrilateral is a polygon of four sidesExamples: M O D E O R L V OA DE RP M E You can name a quadrilateral by its vertices. The order of vertices is veryimportant. You read or write the four letters clockwise or counterclockwise.Examples: R OYou can name a quadrilateral the right as:ROSE or OSER or SERO or EROS E SOr RESO or ESOR or SORE L KThe name of this quadrilateral can be: MKLMN or LMNK or MNKL or NKLMOr NMLK or MLKN or LKNM or KNML NTry this outA. Which of the following is a quadrilateral?1) 3) 5) 3

2) 4) 6)7) 9)8) 10)B. Name the following quadrilaterals.1. 2. C H O O D L E P E P3. 4. B O HA T S 6. M5. A B A LR YY 4

7. M A 8. A B Y ND C 10. S9. O R PRSE QC. Identify an name 10 quadrilaterals in the figure:AB C D EEE hhhdhjhsjsjj S T F U VGH I W K Y NO MSP R Q 5

Lesson 2 Parts of a QuadrilateralA quadrilateral has the following parts:4 sides4 vertices4 angles2 diagonalsYou can take a look on quadrilateral LOVE L O V• the sides are: LO , OV , EV , LE• the vertices are: L, O, V, E• the angles are ∠L , ∠O , ∠V , ∠E E• the diagonals are LV & OEdiagonals are segments joining opposite vertices. The vertices E and O; L and V are opposite vertices. Vertices L and O, Oand V, V and E, E and L are consecutive vertices. Two sides with a common vertex like LO and OV are consecutive sides.So, OV and VE , VE and EL , EL and LO are other pairs of consecutive sides.On the otherhand, LO and EV , OV and LE are opposite sides. Two angles with a common side like ∠L and ∠O are consecutive angles,the others are ∠O and ∠V , ∠V and ∠E , ∠E and ∠L , on the other hand, ∠Eand ∠O ; ∠L and ∠V are opposite angles.Try this out C O A LA. Using quadrilateral COLA, identify:1. two pairs of opposite vertices2. two pairs of opposite angles3. two pairs of opposite sides4. one pair of diagonals5. four pairs of consecutive vertices 6

6. four pairs of consecutive angles7. four pairs of consecutive sidesUsing the figure at the right: P O E8. four sides of quadrilateral POEM M Q9. four angles of quadrilateral POEM R10. two diagonals that can be drawn in quadrilateral POEMRefer to the figure at the right. Fill the blanks:1. ______ the vertex opposite S P2. ______ is the opposite side of PS3. ∠Q is opposite _________4. PQ and QR are _______ sides. S5. The diagonals that can be drawn are QS and ____.Given the figure: B C AD6. BC and CD are ___________ sides.7. B and C are consecutive vertices, B and _____ are also consecutive vertices.8. AC and _____ are the diagonals of quadrilateral ABCD.9. ∠A and ∠D are ______ angles.10. ∠A and _____ are opposite angles. 7

C. Choose the letter of the correct answer. Use quadrilateral D E T H1. How many diagonals has quadrilateral DETH? D E a. one b. twoc. three d. four O2. The sides of quadrilateral DETH areDH , HT , TE and ______a. HE b. DT c. DE d. DO3. ∠LH is opposite of what angle?a. ∠E b. ∠D c. ∠T d. ∠O H I4. The opposite side of ET is _________a. DO b. DE c. HT d. DH5. The diagonals of quadrilateral DETH are _________a. DO & OT b. HE & DT c. DH & ET d. DE & HTGiven: quadrilateral PQRS Q R S6. How many vertices are there in Quadrilateral PQRS? a. one b. twoc. three d. four7. A pair of consecutive sides is PQ and ______a. QR b. RS c. PR d. QS8. A pair of opposite vertices is P and ______a. R b. Q c. S d. M P9. How many pairs of opposite ∠s has quadrilateral PQRS? a. one b. twoc. three d. four10.How many pairs of consecutive sides has quadrilateral PQRS? a. one b. twoc. three d. four 8

Lesson 3 Parallels & PerpendicularsKinds of Quadrilaterals: QuadrilateralTrapezium Trapezoid Parallelogram As you can see the diagram of the different kinds of quadrilaterals, youcan notice the characteristics of the sides of each quadrilateral. Before you proceed to the definition of each quadrilateral, you must knowfirst the meaning of the following:i. parallel lines two lines are parallel if they are coplanar and they do not meet.ii. perpendicular lines two lines are perpendicular if they intersect and form a right angle. 9

Examples: B A two lines are parallel if they are coplanar1. D and they do not meet. C2. l1 is parallel to l2. In symbol l1 // l23. L2 L1 R X If XY intersects RS at O and ∠XOS is a right angle, then XY is perpendicular O S to RS , in symbol: XY ⊥ RS Y4. If ∠MAN is a right angle, then AM ⊥ AN A N 10

Try this outA. Which of the following seem to be parallel? Write yes if it is and no if not.1. 6.2. 7.3. 8.4. 9.5. 10. 11