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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