CHAPTER 4. QUADRILATERALS PARALLELOGRAM AND ITS PROPERTIES By: SAMUEL M. GIER

Define the following: Midpoint of a segment ( a point on the segment that divides the segment into two congruent parts) Congruent segments (are two segments whose measures are equal ) Bisector of an angle ( a ray that divides an angle into two congruent measures)

When are two triangles congruent? If two triangles are congruent, how many pairs of congruent parts can be shown? Name these. CORRESPONDING SIDES FG  XB GH  BM FH  XM CORRESPONDING ANGLES  F   X  G   B  H   M

What are some ways to prove congruent triangles? SSS Congruence Postulate SAS Congruence Postulate ASA Congruence Postulate SAA Congruence Theorem Congruence for Right Triangles Hyl Congruence Theorem HyA congruence Theorem LL Congruence Theorem LA Congruence Theorem

Can the two triangles be proved congruent? If so, what postulate can be used? SSS Congruence Postulate

Can the two triangles be proved congruent? If so, what postulate can be used? SAS Congruence Postulate

Can the two triangles be proved congruent? If so, what postulate can be used? ASA Congruence Postulate

What are some general properties of a parallelogram? The opposite sides are both parallel and congruent. CA R E CA // RE; CA  RE CE // RA ; CE  RA

In the given parallelogram FACE, what does the segment connecting opposite vertices represent? F A F A M E C E C

THE DIAGONALS OF A PARALLELOGRAM OBJECTIVES: 1.To show that the diagonals of a parallelogram bisect each other. 2. To solve problems involving diagonals of a parallelogram.

CLASS ACTIVITY PROCEDURE PROCEDURE 1. Draw and cutout four parallelograms. Construct their diagonals. Let the name of the parallelograms be FACE with the diagonals intersecting at point M. Construct their diagonals. Let the name of the parallelograms be FACE with the diagonals intersecting at point M. 2.With a ruler, measure the distance from the vertex to the point of intersection of the two diagonals. 3.Record your observation.

Data ( Group 1 ) FMCMAMEM Parallelogram 1 Parallelogram 2(square) Parallelogram 3(rectangle) Parallelogram 4(rhombus)

CRITICAL THINKING 1. Compare: FM and CM ; AM and EM. 2. Make a conjecture about the diagonals of a parallelogram FA C E M

Guide Questions 1. In your activity, what can be said about the length of FM compare to the length of CM? How about the length of EM compare to the length of AM? 2. What segment that bisects FC? 3. What segment that bisects AE? 4. What can be said about the diagonals of a parallelogram?

Rubric for the Group Activity Criteria done after 10 minutes 2. answered the 3 follow up questions accurately 3. generalization shows under- standing of the lesson 4. teamwork is evident If all the criteria are met If only 3 of the criteria are met If only 2 of the criteria are met If only 1 or none of the criteria is met

THEOREM THE DIAGONALS OF A PARALLELOGRAM BISECT EACH OTHER. THE DIAGONALS OF A PARALLELOGRAM BISECT EACH OTHER.

Formal proof STATEMENT 1.Parallelogram FACE, with diagonals FC and AE. 2.FA  CE REASON 1.Given 2. Opposite sides of a //gram are congruent. GIVEN: Parallelogram FACE with diagonals FC and AE PROVE: FM  CM ; AM  EM F A C E M PROOF:

Formal proof GIVEN: Parallelogram FACE with diagonals FC and AE PROVE: FM  CM ; AM  EM F A C E M PROOF: STATEMENT 3. FA// EC ;FE // AC 4.  1   4;  2  3 5.  FMA   CME 6. FM  CM AM  EM REASON 3. Definition of//gram 4. If 2 // lines are cut by a transversal, the alternate interior angles are congruent. 5. ASA Congruence 6. CPCTC

EXERCISES: In the given figure, AD and BC are diagonals of //gram ABCD. A B C D O 1.AD = 10 cm, how long is BC? Ans.( 10 cm ) 2. If AB is 30 cm, how long is DC? Ans. ( 30 cm )

EXERCISES: In the given figure, AD and BC are diagonals of //gram ABCD. A B C D O 3. If AO = 15 cm, how long is CO? Ans.( 15 cm ) 4. If DO is 18 cm, how long is BO? Ans. ( 18 cm )

EXERCISES 5. GIVEN: BS = 9x – 4 TS = 7x + 2 FIND : BT SOLUTION: Hence, BS = TS 9x – 4 = 7x +2 9X- 7X = X = 6 X = 3 BS = 23, TS = 23 Therefore, BT = 46 BATH is a parallelogram S B A TH

EXERCISES 6. GIVEN: HS = 5x – 6 AS = 4x + 1 FIND : HA SOLUTION: Hence, HS = AS 5x – 6 = 4x +1 5X- 4X = X = 7 HS = 29; AS = 29 Therefore, HA = 58 BATH is a parallelogram S B A TH

EXERCISES: In the given figure, AD and BC are diagonals of //gram ABCD. A B C D O 7. If AO= (3x-2)cm and CO= (x+8)cm, how long is AC? Ans.( 13 cm ) 8. If DB is 18 cm, how long is BO? Ans. ( 9 cm )

GENERALIZATION WHAT CAN BE SAID ABOUT THE DIAGONALS OF A PARALLELOGRAM?

THEOREM THE DIAGONALS OF A PARALLELOGRAM BISECT EACH OTHER.

VALUING LO E V I How do you relate this property of a parallelogram in our life? What moral lessons we can get out of this topic? FAIRNESS IN DEALING WITH OTHERS.

EVALUATION: 1.If RS + EO = 18 cm and ST = 5 cm, what is ET? 2.If RS + EO = 18 cm and ST = 5 cm, what is RS? 3.If RS = 2x-5 and RT =4, find x and the lengths of RS and ST. RO ES T GIVEN: Parallelogram ROSE with diagonals intersecting at point T.

Agreement Answer Test Yourself nos. 16 – 19, page 128. Geometry Textbook