The document discusses properties of parallelograms, including that the diagonals of a parallelogram bisect each other. It provides definitions of key terms like midpoint and bisector. It also discusses different ways to prove triangles congruent, such as SSS, SAS, and ASA. Examples are given to illustrate finding missing lengths using properties of parallelograms.

1. CHAPTER 4. QUADRILATERALS
PARALLELOGRAM
AND ITS
PROPERTIES

2. 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)

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

4. What are some ways to proveWhat are some ways to prove
congruent triangles?congruent triangles?
SSS CongruenceSSS Congruence
PostulatePostulate
SAS CongruenceSAS Congruence
PostulatePostulate
ASA CongruenceASA Congruence
PostulatePostulate
SAA CongruenceSAA Congruence
TheoremTheorem
Congruence for RightCongruence for Right
TrianglesTriangles
Hyl CongruenceHyl Congruence
TheoremTheorem
HyA congruenceHyA congruence
TheoremTheorem
LL CongruenceLL Congruence
TheoremTheorem
LA CongruenceLA Congruence
TheoremTheorem

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

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

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

8. What are some general propertiesWhat are some general properties
of a parallelogram?of a parallelogram?
The opposite sides are both parallelThe opposite sides are both parallel
and congruent.and congruent.
C A
RE
CA // RE; CA ≅ RE
CE // RA ; CE ≅ RA

10. In the given parallelogram FACE,In the given parallelogram FACE,
what does the segment connectingwhat does the segment connecting
opposite vertices represent?opposite vertices represent?
F AF A
MM
E CE C

11. THE DIAGONALS OF ATHE DIAGONALS OF A
PARALLELOGRAMPARALLELOGRAM
OBJECTIVES:OBJECTIVES:
1.To show that the diagonals of a1.To show that the diagonals of a
parallelogram bisect each other.parallelogram bisect each other.
2. To solve problems involving2. To solve problems involving
diagonals of a parallelogram.diagonals of a parallelogram.

12. CLASS ACTIVITYCLASS ACTIVITY
 PROCEDUREPROCEDURE
1.1. Draw and cutout four parallelograms.Draw and cutout four parallelograms.
Construct their diagonals. Let the nameConstruct their diagonals. Let the name
of the parallelograms beof the parallelograms be FACEFACE with thewith the
diagonals intersecting at pointdiagonals intersecting at point MM..
2.2. With a ruler, measure the distance fromWith a ruler, measure the distance from
the vertex to the point of intersection ofthe vertex to the point of intersection of
the two diagonals.the two diagonals.
3.3. Record your observation.Record your observation.

13. Data ( Group 1 )Data ( Group 1 )
FMFM CMCM AMAM EMEM
ParallelogramParallelogram
11
ParallelogramParallelogram
2(2(squaresquare))
ParallelogramParallelogram
3(3(rectanglerectangle))
ParallelogramParallelogram
4(4(rhombusrhombus))

14. 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 diagonalsMake a conjecture about the diagonals
of a parallelogramof a parallelogram
F A
CE
M

15. Guide QuestionsGuide Questions
1.1. In your activity, what can be said aboutIn your activity, what can be said about
the length of FM compare to the lengththe length of FM compare to the length
of CM? How about the length of EMof 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 ofWhat can be said about the diagonals of
a parallelogram?a parallelogram?

16. THEOREMTHEOREM
THE DIAGONALS OF ATHE DIAGONALS OF A
PARALLELOGRAM BISECTPARALLELOGRAM BISECT
EACH OTHER.EACH OTHER.

17. 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
CE
M
1 2
3 4
PROOF:

18. Formal proof
GIVEN: Parallelogram FACE with
diagonals FC and AE
PROVE:
FM ≅ CM ; AM ≅ EM
F A
CE
M
1 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

19. 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 )

20. 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 )

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

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

23. 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 )

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

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

26. 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.

27. 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.

28. Ysni Ismaili
VII-5