CHAPTER 4. QUADRILATERALS

PARALLELOGRAM AND ITS

PROPERTIES

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?When are two triangles congruent?If two triangles are congruent, If two triangles are congruent, how many pairs of congruent how many pairs of congruent parts can be shown? parts can be shown?

Name these.Name these.

CORRESPONDING SIDESFG ≅ XB GH ≅ BMFH ≅ XM

CORRESPONDING ANGLES∠ F ≅ ∠X∠ G ≅ ∠B∠ H ≅ ∠M

What are some ways to prove What are some ways to prove congruent triangles?congruent triangles?

SSS Congruence SSS Congruence PostulatePostulateSAS Congruence SAS Congruence PostulatePostulateASA Congruence ASA Congruence PostulatePostulateSAA Congruence SAA Congruence TheoremTheorem

Congruence for Right Congruence for Right TrianglesTrianglesHyl Congruence Hyl Congruence TheoremTheoremHyA congruence HyA congruence TheoremTheoremLL Congruence LL Congruence TheoremTheoremLA Congruence LA Congruence TheoremTheorem

Can the two triangles be proved Can the two triangles be proved congruent? If so, what postulate congruent? If so, what postulate

can be used?can be used?

SSS Congruence

Postulate

Can the two triangles be proved Can the two triangles be proved congruent? If so, what postulate congruent? If so, what postulate

can be used?can be used?

SAS Congruence

Postulate

Can the two triangles be proved Can the two triangles be proved congruent? If so, what postulate congruent? If so, what postulate

can be used?can be used?

ASA Congruence

Postulate

What are some general properties What are some general properties of a parallelogram?of a parallelogram?

The opposite sides are both parallel The opposite sides are both parallel and congruent. and congruent.

C A

RE

CA // RE; CA ≅ RE

CE // RA ; CE ≅ RA

In the given parallelogram FACE, In the given parallelogram FACE, what does the segment connecting what does the segment connecting

opposite vertices represent? opposite vertices represent?

F AF A

MM

E CE C

THE DIAGONALS OF A THE DIAGONALS OF A PARALLELOGRAMPARALLELOGRAM

OBJECTIVES:OBJECTIVES:1.To show that the diagonals of a 1.To show that the diagonals of a parallelogram bisect each other.parallelogram bisect each other.2. To solve problems involving 2. To solve problems involving diagonals of a parallelogram. diagonals of a parallelogram.

CLASS ACTIVITYCLASS ACTIVITY PROCEDUREPROCEDURE1.1. Draw and cutout four parallelograms.Draw and cutout four parallelograms. Construct their diagonals. Let the name Construct their diagonals. Let the name

of the parallelograms be of the parallelograms be FACEFACE with the with the diagonals intersecting at point diagonals intersecting at point MM..

2.2. With a ruler, measure the distance from With a ruler, measure the distance from the vertex to the point of intersection of the vertex to the point of intersection of the two diagonals. the two diagonals.

3.3. Record your observation.Record your observation.

Data ( Group 1 )Data ( Group 1 )

FMFM CMCM AMAM EMEM

Parallelogram Parallelogram 11

Parallelogram Parallelogram 2(2(squaresquare))

Parallelogram Parallelogram 3(3(rectanglerectangle))

Parallelogram Parallelogram 4(4(rhombusrhombus))

CRITICAL THINKINGCRITICAL THINKING

1.1. Compare: FM and CM ; AM and EM.Compare: FM and CM ; AM and EM.2.2. Make a conjecture about the diagonals Make a conjecture about the diagonals

of a parallelogramof a parallelogram

F A

CE

M

Guide QuestionsGuide Questions1.1. In your activity, what can be said about In your activity, what can be said about

the length of FM compare to the length the length of FM compare to the length of CM? How about the length of EM of CM? How about the length of EM compare to the length of AM? compare to the length of AM?

2.2. What segment that bisects FC?What segment that bisects FC?3.3. What segment that bisects AE?What segment that bisects AE?4.4. What can be said about the diagonals of What can be said about the diagonals of

a parallelogram?a parallelogram?

THEOREMTHEOREM

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

Formal proof

STATEMENT1. Parallelogram

FACE, with diagonals FC and AE.

2. FA ≅ CE

REASON1. Given

2. Opposite sides of a //gram are congruent.

GIVEN: Parallelogram FACE with diagonals FC and AE

PROVE: FM ≅ CM ; AM ≅ EM

F A

CE

M1 2

3 4

PROOF:

Formal proofGIVEN: Parallelogram FACE with diagonals FC and AE

PROVE: FM ≅ CM ; AM ≅ EM

F A

CE

M1 2

3 4

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

CD

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 )

EXERCISES5. GIVEN: BS = 9x – 4 TS = 7x + 2 FIND : BTSOLUTION:Hence, BS = TS9x – 4 = 7x +29X- 7X = 2 + 4 2X = 6 X = 3BS = 23, TS = 23Therefore, BT = 46

BATH is a parallelogram

S

B A

TH

EXERCISES

6. GIVEN: HS = 5x – 6 AS = 4x + 1 FIND : HASOLUTION:Hence, HS = AS5x – 6 = 4x +15X- 4X = 1 + 6 X = 7 HS = 29; AS = 29Therefore, 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

CD

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

L O

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.

R O

E S

T

GIVEN:Parallelogram ROSE with diagonals intersecting at point T.

Ysni IsmailiVII-5