1 Marks Quetions
1. A quadrilateral ABCD is a parallelogram if
(a) AB = CD
(b) ABBC
(c)
(d) AB = AD
Ans. (c)
2. In figure, ABCD and AEFG are both parallelogram if
(a)
(b)
(c)
(d)
Ans. (c)
3. In a square ABCD, the diagonals AC and BD bisects at O. Then is
(a) acute angled
(b) obtuse angled
(c) equilateral
(d) right angled
Ans. (d) right angled
4. ABCD is a rhombus. If
(a)
(b)
(c)
(d)
Ans. (c)
5. In fig ABCD is a parallelogram. If and
then
is
Ans.
6. If the diagonals of a quadrilateral bisect each other, then the quadrilateral must be.
(a) Square
(b) Parallelogram
(c) Rhombus
(d) Rectangle
Ans. (b) Parallelogram
7. The diagonal AC and BD of quadrilateral ABCD are equal and are perpendicular bisector of each other then quadrilateral ABCD is a
(a) Kite
(b) Square
(c) Trapezium
(d) Rectangle
Ans. (b) Square
8. The quadrilateral formed by joining the mid points of the sides of a quadrilateral ABCD taken in order, is a rectangle if
(a) ABCD is a parallelogram
(b) ABCD is a rut angle
(c) Diagonals AC and BD are perpendicular
(d) AC=BD
Ans. (a) ABCD is a parallelogram
9. In the fig ABCD is a Parallelogram. The values of and
are
(a) 30, 35
(b) 45, 30
(c) 45, 45
(d) 55, 35
Ans. (b) 45, 30
10. In fig if DE=8 cm and D is the mid-Point of AB, then the true statement is
(a) AB=AC
(b) DE||BC
(c) E is not mid-Point of AC
(d) DEBC
Ans. (c) E is not mid-Point of AC
11. The sides of a quadrilateral extended in order to form exterior angler. The sum of these exterior angle is
(a)
(b)
(c)
(d)
Ans. (d)
12. ABCD is rhombus with The measure of
is
(a)
(b)
(c)
(d)
Ans. b)
13. In fig D is mid-point of AB and DEBC then AE is equal to
(a) AD
(b) EC
(c) DB
(d) BC
Ans. (b) EC
14. In fig D and E are mid-points of AB and AC respectively. The length of DE is
(a) 8.2 cm
(b) 5.1 cm
(c) 4.9 cm
(d) 4.1 cm
Ans. (d) 4.1 cm
15. A diagonal of a parallelogram divides it into
(a) two congruent triangles
(b) two similes triangles
(c) two equilateral triangles
(d) none of these
Ans. (a) two congruent triangles
16. A quadrilateral is a _________, if its opposite sides are equal:
(a) Kite
(b) trapezium
(c) cyclic quadrilateral
(d) parallelogram
Ans. (d) parallelogram
17. In the adjoining Fig. AB = AC. CD||BA and AD is the bisector of prove that
(a) and
Ans. In
Now
Also CD||BA Given)
is a parallelogram
(ii) ABCD is a parallelogram
18. Which of the following is not a parallelogram?
(a) Rhombus
(b) Square
(c) Trapezium
(d) Rectangle
Ans. (c) Trapezium
19. The sum of all the four angles of a quadrilateral is
(a) 1800
(b) 3600
(c) 2700
(d) 900
Ans. (b) 3600
20. In Fig ABCD is a rectangle P and Q are mid-points of AD and DC respectively. Then length of PQ is
(a)5 cm
(b) 4 cm
(c) 2.5 cm
(d) 2 cm
Ans. (c) 2.5 cm
21. In Fig ABCD is a rhombus. Diagonals AC and BD intersect at O. E and F are mid points of AO and BO respectively. If AC = 16 cm and BD = 12 cm then EF is
(a)10 cm
(b) 5 cm
(c) 8 cm
(d) 6 cm
Ans. (b) 5 cm
2 Marks Quetions
1. The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all angles of the quadrilateral
Ans. Let in quadrilateral ABCD, A =
B =
C =
and
D =
Since, sum of all the angles of a quadrilateral =
A +
B +
C +
D =
Now A =
B =
C =
And D =
Hence angles of given quadrilateral are and
2. If the diagonals of a parallelogram are equal, show that it is a rectangle.
Ans. Given: ABCD is a parallelogram with diagonal AC = diagonal BD
To prove: ABCD is a rectangle.
Proof: In triangles ABC and ABD,
AB = AB
AC = BD
AD = BC [opp. Sides of a gm]
ABC
BAD
DAB =
CBA
But DAB +
CBA =
……….(ii)
[ AD
BC and AB cuts them, the sum of the interior angles of the same side of transversal is
]
From eq. (i) and (ii),
DAB =
CBA =
Hence ABCD is a rectangle.
3. Diagonal AC of a parallelogram ABCD bisects A (See figure). Show that:
(i) It bisects C also.
(ii) ABCD is a rhombus.
Ans. Diagonal AC bisects A of the parallelogram ABCD.
(i) Since AB DC and AC intersects them.
1 =
3
Similarly 2 =
4 ……….(ii)
But 1 =
2
3 =
4
Thus AC bisects C.
(ii) 2 =
3 =
4 =
1
AD = CD
AB = CD = AD = BC
Hence ABCD is a rhombus.
4. ABCD is a parallelogram and AP and CQ are the perpendiculars from vertices A and C on its diagonal BD (See figure). Show that:
(i) APB
CQD
(ii) AP = CQ
Ans. Given: ABCD is a parallelogram. AP BD and CQ
BD
To prove: (i) APB
CQD (ii) AP = CQ
Proof: (i) InAPB and
CQD,
1 =
2
AB = CD
APB =
CQD =
APB
CQD
(ii) Since APB
CQD
AP = CQ
5. ABCD is a quadrilateral in which P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively (See figure). AC is a diagonal. Show that:
(i) SR AC and SR =
AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
Ans. In ABC, P is the mid-point of AB and Q is the mid-point of BC.
Then PQ AC and PQ =
AC
(i) In ACD, R is the mid-point of CD and S is the mid-point of AD.
Then SR AC and SR =
AC
(ii) Since PQ = AC and SR =
AC
Therefore, PQ = SR
(iii) Since PQ AC and SR
AC
Therefore, PQ SR
Now PQ = SR and PQ SR
Therefore, PQRS is a parallelogram.
6. The angles of a quadrilateral are in the ratio 3:5:9:13. Find all the angles of the quadrilateral.
Ans. Suppose angles of quadrilateral ABCD are 3x, 5x, 9x, and 13x
[sum of angles of a quadrilateral is
]
7. Show that each angle of a rectangle is a right angle.
Ans. We know that rectangle is a parallelogram whose one angle is right angle.
Let ABCD be a rectangle.
To prove
Proof: and AB is transversal
8. A transversal cuts two parallel lines prove that the bisectors of the interior angles enclose a rectangle.
Ans. and EF cuts them at P and R.
9. Prove that diagonals of a rectangle are equal in length.
Ans. ABCD is a rectangle and AC and BD are diagonals.
To prove AC = BD
Proof:
AD = BC
AB = AB common
10. If each pair of opposite sides of a quadrilateral is equal, then prove that it is a parallelogram.
Ans. Given A quadrilateral ABCD in which AB = DC and AD = BC
To prove: ABCD is a parallelogram
Construction: Join AC
Proof: In and
AD=BC (Given)
AB=DC
AC=AC
is a parallelogram.
11.
Ans. ABCD is a parallelogram. The diagonals of a parallelogram bisect bisect each other
But
Or OX=OY
Now in quadrilateral AYCX, the diagonals AC and XY bisect each other
is a parallelogram.
In fig ABCD is a parallelogram and x, y are the points on the diagonal BD such that Dx<By show that AYCX is a parallelogram.
12. Show that the line segments joining the mid points of opposite sides of a quadrilateral bisect each other.
Ans. Given ABCD is quadrilateral E, F, G, H are mid points of the side AB, BC, CD and DA respectively
To prove: EG and HF bisect each other.
In , E is mid-point of AB and F is mid-point of BC
And
Similarly, and
From (i) and (ii), and
is a parallelogram and EG and HF are its diagonals
Diagonals of a parallelogram bisect each other
Thus, EG and HF bisect each other.
13. ABCD is a rhombus show that diagonal AC bisects as well as
and diagonal BD bisects
as well as
Ans. ABCD is a rhombus
In and
And
Hence AC bisects as well as
Similarly, by joining B to D, we can prove that
Hence BD bisects as well as
14. In fig AD is a median of is mid-Point of AD.BE produced meet AC at F. Show that
Ans. Let M is mid-Point of CF Join DM
In is mid- Point of AD and
is mid-point of AM
FM=MC
Hence Proved.
15. Prove that a quadrilateral is a parallelogram if the diagonals bisect each other.
Ans. ABCD is a quadrilateral in which diagonals AC and BD intersect each other at O
In and
And [Vertically apposite angle
But this is Pair of alternate interior angles
Similarly AD||BC
Quadrilateral ABCD is a Parallelogram.
16. In fig ABCD is a Parallelogram. AP and CQ are Perpendiculars from the Vertices A and C on diagonal BD.
Show that
(i)
(ii)
Ans. (I) in and
AB=DC
And
(II) (By C.P.C.T)
17. ABCD is a Parallelogram E and F are the mid-Points of BC and AD respectively. Show that the segments BF and DE trisect the diagonal AC.
Ans. FD||BE and FD=BE
BEDF Is a Parallelogram
EG||BH and E is the mid-Point of BC
G is the mid-point of HC
Or HG=GC…………..(i)
Similarly AH=HG………….(ii)
From (i) and (ii) we get
AH=HG=GC
Thus the segments BF and DE bisects the diagonal AC.
18. Prove that if each pair of apposite angles of a quadrilateral is equal, then it is a parallelogram.
Ans. Given: ABCD is a quadrilateral in which and
To Prove: ABCD is a parallelogram
Proof:
In quadrilateral. ABCD
These are sum of interior angles on the same side of transversal
and
ABCD is a parallelogram.
19. In Fig. ABCD is a trapezium in which AB||DC E is the mid-point of AD. A line through E is parallel to AB show that bisects the side BC
Ans. Join AC
In
E is mid-point of AD and EO||DC
O is mid point of AC [A line segment joining the midpoint of one side of a
parallel to second side and bisect the third side]
In
O is mid point of AC
OF||AB F is mid point of BC
Bisect BC
20. In Fig. ABCD is a parallelogram in which X and Y are the mid-points of the sides DC and AB respectively. Prove that AXCY is a parallelogram
Ans. In the given fig
ABCD is a parallelogram
AB||CD and AB = CD
And
And
is a parallelogram
21. The angles of quadrilateral are in the ratio 3:5:10:12 Find all the angles of the quadrilateral.
Ans. Suppose angles of quadrilaterals are
In a quadrilateral
30x=360
22. In fig D is mid-points of AB. P is on AC such that and DE||BP show that
Ans. In ABP
D is mid points of AB and DE||BPE is midpoint of AP
AE = EP also PC =
AP
2PC = AP
2PC = 2AEPC = AE
AE = PE = PC
AC = AE + EP + PC
AC = AE + AE + AEAE =
AC
Hence Proved.
23. Prove that the bisectors of the angles of a Parallelogram enclose a rectangle. It is given that adjacent sides of the parallelogram are unequal.
Ans. ABCD is a parallelogram
Or [Sum of angle of a
]
Similarly, and
. Thus each angle of quadrilateral PQRS is
Hence PQRS is a rectangle.
24. Prove that a quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal
Ans. Given: ABCD is a quadrilateral in which AB||DC and BC||AD.
To Prove: ABCD is a parallelogram
Construction: Join AC and BD intersect each other at O.
Proof: [By AAA
Because AO=OC
And BO=OD ABCD is a parallelogram
Diagonals of a parallelogram bisect each other.
3 Marks Quetions
1. Show that is diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Ans. Given: Let ABCD is a quadrilateral.
Let its diagonal AC and BD bisect each other at right angle at point O.
OA = OC, OB = OD
AndAOB =
BOC =
COD =
AOD =
To prove: ABCD is a rhombus.
Proof: In AOD and
BOC,
OA = OC
AOD =
BOC
OB = OD
AOD
COB
AD = CB
Again, In AOB and
COD,
OA = OC
AOB =
COD
OB = OD
AOB
COD
AD = CB
Now In AOD and
BOC,
OA = OC
AOB =
BOC
OB = OB
AOB
COB
AB = BC
From eq. (i), (ii) and (iii),
AD = BC = CD = AB
And the diagonals of quadrilateral ABCD bisect each other at right angle.
Therefore, ABCD is a rhombus.
2. Show that the diagonals of a square are equal and bisect each other at right angles.
Ans. Given: ABCD is a square. AC and BD are its diagonals bisect each other at point O.
To prove: AC = BD and AC BD at point O.
Proof: In triangles ABC and BAD,
AB = AB
ABC =
BAD =
BC = AD
ABC
BAD
AC = BD
Now in triangles AOB and AOD,
AO = AO
AB = AD
OB = OD
AOB
AOD
AOB =
AOD
ButAOB +
AOD =
AOB =
AOD =
OA
BD or AC
BD
Hence proved.
3. ABCD is a rhombus. Show that the diagonal AC bisects A as well as
C and diagonal BD bisects
B as well as
D.
Ans. ABCD is a rhombus. Therefore, AB = BC = CD = AD
Let O be the point of bisection of diagonals.
OA = OC and OB = OD
In AOB and
AOD,
OA = OA
AB = AD
OB = OD(diagonals of rhombus bisect each other]
AOB
AOD
OAD =
OAB
OA bisects
A……….(i)
SimilarlyBOC
DOC
OCB =
OCD
OC bisects
C……….(ii)
From eq. (i) and (ii), we can say that diagonal AC bisects A and
C.
Now in AOB and
BOC,
OB = OB
AB = BC
OA = OC(diagonals of rhombus bisect each other]
AOB
COB
OBA =
OBC
OB bisects
B……….(iii)
SimilarlyAOD
COD
ODA =
ODC
BD bisects
D……….(iv)
From eq. (iii) and (iv), we can say that diagonal BD bisects B and
D
4. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (See figure). Show that:
(i) APD
CQB
(ii) AP = CQ
(iii) AQB
CPD
(iv) AQ = CP
(v) APCQ is a parallelogram.
Ans. (i)In APD and
CQB,
DP = BQ
ADP =
QBC[Alternate angles (AD
BC and BD is transversal)]
AD = CB
APD
CQB
(ii) SinceAPD
CQB
AP = CQ
(iii) In AQB and
CPD,
BQ = DP
ABQ =
PDC[Alternate angles (AB
CD and BD is transversal)]
AB = CD
AQB
CPD
(iv) SinceAQB
CPD
AQ = CP
(v) In quadrilateral APCQ,
AP = CQ
AQ = CP
Since opposite sides of quadrilateral APCQ are equal.
Hence APCQ is a parallelogram.
5. ABCD is a rhombus and P, Q, R, S are mid-points of AB, BC, CD and DA respectively. Prove that quadrilateral PQRS is a rectangle.
Ans. Given: P, Q, R and S are the mid-points of respective sides AB, BC, CD and DA of rhombus. PQ, QR, RS and SP are joined.
To prove: PQRS is a rectangle.
Construction: Join A and C.
Proof: In ABC, P is the mid-point of AB and Q is the mid-point of BC.
PQ
AC and PQ =
AC ……….(i)
In ADC, R is the mid-point of CD and S is the mid-point of AD.
SR
AC and SR =
AC……….(ii)
From eq. (i) and (ii),PQ SR and PQ = SR
PQRS is a parallelogram.
Now ABCD is a rhombus.
AB = BC
AB =
BC
PB = BQ
1 =
2
Now in triangles APS and CQR, we have,
AP = CQ
Similarly AS = CR and PS = QR
APS
CQR
3 =
4
Now we have1 +
SPQ +
3 =
And2 +
PQR +
4 =
1 +
SPQ +
3 =
2 +
PQR +
4
Since1 =
2 and
3 =
4
SPQ =
PQR……….(iii)
Now PQRS is a parallelogram
SPQ +
PQR =
……….(iv)
Using eq. (iii) and (iv),
SPQ +
SPQ =
2
SPQ =
SPQ =
Hence PQRS is a rectangle.
6. ABCD is a rectangle and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
Ans. Given: A rectangle ABCD in which P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. PQ, QR, RS and SP are joined.
To prove: PQRS is a rhombus.
Construction: Join AC.
Proof: In ABC, P and Q are the mid-points of sides AB, BC respectively.
PQ
AC and PQ =
AC……….(i)
In ADC, R and S are the mid-points of sides CD, AD respectively.
SR
AC and SR =
AC……….(ii)
From eq. (i) and (ii), PQ SR and PQ = SR……….(iii)
PQRS is a parallelogram.
Now ABCD is a rectangle.
AD = BC
AD =
BC
AS = BQ……….(iv)
In triangles APS and BPQ,
AP = BP
PAS =
PBQ[Each
]
And AS = BQ
APS
BPQ
PS = PQ
From eq. (iii) and (v), we get that PQRS is a parallelogram.
PS = PQ
Two adjacent sides are equal.
Hence, PQRS is a rhombus.
7. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (See figure). Show that the line segments AF and EC trisect the diagonal BD.
Ans. Since E and F are the mid-points of AB and CD respectively.
AE =
AB and CF =
CD……….(i)
But ABCD is a parallelogram.
AB = CD and AB
DC
AB =
CD and AB
DC
AE = FC and AE
FC
AECF is a parallelogram.
FA
CE
FP
CQ
Since the segment drawn through the mid-point of one side of a triangle and parallel to the other side bisects the third side.
In DCQ, F is the mid-point of CD and
FP
CQ
P is the mid-point of DQ.
DP = PQ……….(iii)
Similarly, In ABP, E is the mid-point of AB and
EQ
AP
Q is the mid-point of BP.
BQ = PQ……….(iv)
From eq. (iii) and (iv),
DP = PQ = BQ………(v)
Now BD = BQ + PQ + DP = BQ + BQ + BQ = 3BQ
BQ =
BD……….(vi)
From eq. (v) and (vi),
DP = PQ = BQ = BD
Points P and Q trisects BD.
So AF and CE trisects BD.
8. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D.
Ans. (i) In ABC, M is the mid-point of AB
MD BC
AD = DC
Thus D is the mid-point of AC.
(ii) BC (given) consider AC as a transversal.
1 =
C
1 =
[
C =
]
Thus MD AC.
(iii) In AMD and
CMD,
AD = DC
1 =
2 =
MD = MD
AMD
CMD
AM = CM
Given that M is the mid-point of AB.
AM =
AB……….(ii)
From eq. (i) and (ii),
CM = AM = AB
9. In a parallelogram ABCD, bisectors of adjacent angles A and B intersect each other at P. prove that
Ans. Given ABCD is a parallelogram is and bisectors of intersect each other at P.
To prove
Proof:
But ABCD is a parallelogram and ADBC
Hence Proved
10. In figure diagonal AC of parallelogram ABCD bisects show that
(i) if bisects
ABCD is a rhombus
Ans.(i) ABDC and AC is transversal
(Alternate angles)
And (Alternate angles)
But,
(ii)
AC=AC
11. In figure ABCD is a parallelogram. AX and CY bisects angles A and C. prove that AYCX is a parallelogram.
Ans. Given in a parallelogram AX and CY bisects respectively and we have to show that AYCX in a parallelogram.
…(i)
And
But
By (2) and (3), we get
Also,
From (i), (iv) and (v), we get
But, AB =CD
AB-BY=CD-DX
Or
Ay=CX
But
is a parallelogram
12. Prove that the line segment joining the mid-points of two sides of a triangle is parallel to the third side.
Ans. Given ABC in which E and F are mid points of side AB and AC respectively.
To prove: EF||BC
Construction: Produce EF to D such that EF = FD. Join CD
Proof:
AF=FC
EF=FD
And
AE= BE[is the mid-point]
And
13. Prove that a quadrilateral is a rhombus if its diagonals bisect each other at right angles.
Ans. Given ABCD is a quadrilateral diagonals AC and BD bisect each other at O at right angles
To Prove: ABCD is a rhombus
Proof: diagonals AC and BD bisect each other at O
And
Now In And
Given
And (Given)
(SAS)
(C.P.C.T.)
Similarly, BC=CD, CD=DA and DA=AB,
Hence, ABCD is a rhombus.
14. Prove that the straight line joining the mid points of the diagonals of a trapezium is parallel to the parallel sides.
Ans. Given a trapezium ABCD in which and M,N are the mid Points of the diagonals AC and BD.
We need to prove that
Join CN and let it meet AB at E
Now in and
And
Now in and
are the mid points of the sides AC and CE respectively.
Or
Also
15. In fig is a right angle in
is the mid-point of
intersects BC at E. show that
(i) E is the mid-point of BC
(ii) DEBC
(ii) BD = AD
Ans. Proof:and D is mid points of AC
In and
CE=BE
DE= DE
And
16. ABC is a triangle and through vertices A, B and C lines are drawn parallel to BC, AC and AB respectively intersecting at D, E and F. prove that perimeter of is double the perimeter of
.
Ans. Is a parallelogram
Is a parallelogram
Or
Similarly, ED = 2AB and FD = 2AC
Perimeter of
Perimeter of
= 2AB+2BC+2AC
= 2
= 2 Perimeter of
Hence Proved.
17. In fig ABCD is a quadrilateral P, Q, R and S are the mid Points of the sides AB, BC, CD and DA, AC is diagonal. Show that
(i) SR||AC
(ii) PQ=SR
(iii) PQRS is a parallelogram
(iv) PR and SQ bisect each other
Ans. In ABC, P and Q are the mid-points of the sides AB and BC respectively
(i) PQ||AC and PQ=
AC
(ii) Similarly SR||AC and SR=AC
PQ||SR and PQ=SR
(iii) Hence PQRS is a Parallelogram.
(iv) PR and SQ bisect each other.
18. In are respectively the mid-Points of sides AB,DC and CA. show that
is divided into four congruent triangles by Joining D,E,F.
Ans. D and E are mid-Points of sides AB and BC of ABC
DE||AC {
A line segment joining the mid-Point of any two sides of a triangle parallel to third side}
Similarly, DF||BC and EF||AB
ADEF, BDEF and DFCE are all Parallelograms.
DE is diagonal of Parallelogram BDFE
Similarly, DAF
FED
And EFC
FED
So all triangles are congruent
19. ABCD is a Parallelogram is which P and Q are mid-points of opposite sides AB and CD. If AQ intersect DP at S BQ intersects CP at R, show that
(i) APCQ is a Parallelogram
(ii) DPBQ is a parallelogram
(iv) PSQR is a parallelogram
Ans. (i) In quadrilateral APCQ
AP||QC [AB||CD]……..(i)
AP=AB, CQ=
CD (Given)
Also AB= CD
So AP=QC……….(ii)
Therefore, APCQ is a parallelogram
(ii) Similarly, quadrilateral DPBQ is a Parallelogram because DQ||PB and DQ=PB
(iii) In quadrilateral PSQR,
SP||QR
Similarly, SQ||PR
So. PSQR is also parallelogram.
20.are three parallel lines intersected by transversals P and q such that
and
cut off equal intercepts AB and BC on P In fig Show that
cut off equal intercepts DE and EF on q also.
Ans. In fig are 3 parallel lines intersected by two transversal P and Q.
To Prove DE=EF
Proof: In
B is mid-point of AC
And BG||CF
G is mid-point of AF
Now In
G is mid-point of AF and GE || AD
E is mid-point of FD
DE=EF
Hence Proved.
21. ABCD is a parallelogram in which E is mid-point of AD. DF||EB meeting AB produced at F and BC at L prove that DF = 2DL
Ans. In
is mid-point of AD (Given)
BE||DF (Given)
By converse of mid-point theorem B is mid-point of AF
ABCD is parallelogram
From (i) and (ii)
CD = BF
Consider and
DC = FB
DL=LF
DF=2DL
22. PQRS is a rhombus if find
Ans.
Let
In we have RS=RQ
In
23. ABCD is a trapezium in which AB||CD and AD = BC show that
(i)
(ii)
(iii)
Ans. Produce AB and Draw a line Parallel to DA meeting at E
AD||EC
….(i) [Sum of interior angles on the some side of transversal is
]
In
BC=CE (given)
…..(2) [in a
equal side to opposite angles are equal]
……(3)
By (i) and (3)
(i)
(ii)
(iii) In ABC and
BAD
AB=AB
AD=BC
24. Show that diagonals of a rhombus are perpendicular to each other.
Ans. Given: A rhombus ABCD whose diagonals AC and BD intersect at a Point O
To Prove:
Proof: clearly ABCD is a Parallelogram in which
AB=BC=CD=DA
We know that diagonals of a Parallelogram bisect each other
OA=OC and OB=OD
Now in BOC and
DOC, we have
OB=OD
BC=DC
OC=OC
BOC
DOC
But
Similarly,
Hence diagonals of a rhombus bisect each other at
25. Prove that the diagonals of a rhombus bisect each other at right angles
Ans. We are given a rhombus ABCD whose diagonals AC and BD intersect each other at O.
We need to prove that OA=OC, OB=OD and
In and
AB=CD
And
AOB
COD
OA=OC
And OB=OD
Also in AOB and
COB
OA=OC
AB=CB
And OB=OB
AOB
COB
But
26. In fig ABCD is a trapezium in which AB||DC and AD=BC. Show that
Ans. To show that
Draw CP||DA meeting AB at P
AP||DC and CP||DA
APCD is a parallelogram
Again in CPB
CP=CB [BC=AD
But
Also [
APCD is a parallelogram]
Or
= CB
27. In fig ABCD and ABEF are Parallelogram, prove that CDFE is also a parallelogram.
Ans. ABCD is a parallelogram
AB=DC also AB||DC…………..(i)
Also ABEF is a parallelogram
AB=FE and AB||FE……….(ii)
By (i) and (ii)
AB=DC=FE
AB=FE
And AB||DC||FE
AB||FE
CDEF is a parallelogram.
Hence Proved.
4 Marks Quetions
1. ABCD is a rectangle in which diagonal AC bisects A as well as
C. Show that:
(i) ABCD is a square.
(ii) Diagonal BD bisects both B as well as
D.
Ans. ABCD is a rectangle. Therefore AB = DC ……….(i)
And BC = AD
Also A =
B =
C =
D =
(i) In ABC and
ADC
1 =
2 and
3 =
4
[AC bisects A and
C (given)]
AC = AC
ABC
ADC
AB = AD ……….(ii)
From eq. (i) and (ii), AB = BC = CD = AD
Hence ABCD is a square.
(ii) In ABC and
ADC
AB = BA
AD = DC
BD = BD
ABD
CBD
ABD =
CBD
And ADB =
CDB
From eq. (iii) and (iv), it is clear that diagonal BD bisects both B and
D.
2. An ABC and
DEF, AB = DE, AB
DE, BC = EF and BC
EF. Vertices A, B and C are joined to vertices D, E and F respectively (See figure). Show that:
(i) Quadrilateral ABED is a parallelogram.
(ii) Quadrilateral BEFC is a parallelogram.
(iii) AD CF and AD = CF
(iv) Quadrilateral ACFD is a parallelogram.
(v) AC = DF
(vi) ABC
DEF
Ans. (i) In ABC and
DEF
AB = DE
And AB DE
ABED is a parallelogram.
(ii) In ABC and
DEF
BC = EF
And BC EF
BEFC is a parallelogram.
(iii) As ABED is a parallelogram.
AD
BE and AD = BE ……….(i)
Also BEFC is a parallelogram.
CF
BE and CF = BE ……….(ii)
From (i) and (ii), we get
AD
CF and AD = CF
(iv) As AD CF and AD = CF
ACFD is a parallelogram.
(v) As ACFD is a parallelogram.
AC = DF
(vi) InABC and
DEF,
AB = DE
BC = EF
AC = DF
ABC
DEF
3. ABCD is a trapezium, in which AB DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E, parallel to AB intersecting BC at F (See figure). Show that F is the mid-point of BC.
Ans. Let diagonal BD intersect line EF at point P.
In DAB,
E is the mid-point of AD and EP AB [
EF
AB (given) P is the part of EF]
P is the mid-point of other side, BD of
DAB.
Now in BCD,
P is the mid-point of BD and PF DC [
EF
AB (given) and AB
DC (given)]
EF
DC and PF is a part of EF.
F is the mid-point of other side, BC of
BCD.
4. Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisect each other.
Ans. Given: A quadrilateral ABCD in which EG and FH are the line-segments joining the mid-points of opposite sides of a quadrilateral.
To prove: EG and FH bisect each other.
Construction: Join AC, EF, FG, GH and HE.
Proof: In ABC, E and F are the mid-points of respective sides AB and BC.
EF
AC and EF
AC ……….(i)
Similarly, in ADC,
G and H are the mid-points of respective sides CD and AD.
HG
AC and HG
AC ……….(ii)
From eq. (i) and (ii),
EF HG and EF = HG
EFGH is a parallelogram.
Since the diagonals of a parallelogram bisect each other, therefore line segments (i.e. diagonals) EG and FH (of parallelogram EFGH) bisect each other.
5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
Ans. Let ABCD be a quadrilateral in which equal diagonals AC and BD bisect each other at right angle at point O.
We have AC = BD and OA = OC ……….(i)
And OB = OD ……….(ii)
Now OA + OC = OB + OD
OC + OC = OB + OB
2OC = 2OB
OC = OB ……….(iii)
From eq. (i), (ii) and (iii), we get, OA = OB = OC = OD ……….(iv)
Now in AOB and
COD,
OA = OD
AOB =
COD
OB = OC
AOB
DOC
AB = DC
Similarly, BOC
AOD
BC = AD
From eq. (v) and (vi), it is concluded that ABCD is a parallelogram because opposite sides of a quadrilateral are equal.
Now in ABC and
BAD,
AB = BA
BC = AD
AC = BD
ABC
BAD
ABC =
BAD
But ABC +
BAD =
AD
BC and AB is a transversal.
ABC +
ABC =
2
ABC =
ABC =
ABC =
BAD =
……….(ix)
Opposite angles of a parallelogram are equal.
But ABC =
BAD =
ABC =
ADC =
……….(x)
BAD =
BDC =
……….(xi)
From eq. (x) and (xi), we get
ABC =
ADC =
BAD =
BDC =
……….(xii)
Now in AOB and
BOC,
OA = OC
AOB =
BOC =
OB = OB
AOB
COB
AB = BC ……….(xiii)
From eq. (v), (vi) and (xiii), we get,
AB = BC = CD = AD ……….(xiv)
Now, from eq. (xii) and (xiv), we have a quadrilateral whose equal diagonals bisect each other at right angle.
Also sides are equal make an angle of with each other.
ABCD is a square.
6. ABCD is a trapezium in which AB CD and AD = BC (See figure). Show that:
(i) A =
B
(ii) C =
D
(iii) ABC
BAD
(iv) Diagonal AC = Diagonal BD
Ans. Given: ABCD is a trapezium.
AB CD and AD = BC
To prove: (i) A =
B
(ii) C =
D
(iii) ABC
BAD
(iv) Diag. AC = Diag. BD
Construction: Draw CE AD and extend AB to intersect CE at E.
Proof: (i) As AECD is a parallelogram.
AD = EC
But AD = BC
BC = EC
3 =
4
Now 1 +
4 =
And 2 +
3 =
1 +
4 =
2 +
3
1 =
2 [
3 =
4 ]
A =
B
(ii) 3 =
C
And D =
4
But 3 =
4 [
BCE is an isosceles triangle]
C =
D
(iii) In ABC and
BAD,
AB = AB
1 =
2
AD = BC
ABC
BAD
AC = BD
7. Prove that if the diagonals of a quadrilateral are equal and bisect each other at right angles then it is a square.
Ans. Given in a quadrilateral ABCD, AC = BD, AO = OC and BO = OD and
To prove: ABCD is a square.
Proof:
OA=OC
OB=OD
And
But these are alternate angles
ABCD is a parallelogram whose diagonals bisects each other at right angles
Again in
AB=BC
AD=AB
And BD=CA
These are alternate angles of these same side of transversal
Hence ABCD is a square.
8. Prove that in a triangle, the line segment joining the mid points of any two sides is parallel to the third side.
Ans. Given: A in which D and E are mid-points of the side AB and AC respectively
To Prove:
Construction: Draw
Proof: In and
AE=CE
And
DE=FE
But DA = DB
DB = FC
Now DBFC
DBCF is a parallelogram
DE
BC
Also DE = EF = BC
9. ABCD is a rhombus and P, Q, R, and S are the mid-Points of the sides AB, BC, CD and DA respectively. Show that quadrilateral PQRS is a rectangle.
Ans. Join AC and BD which intersect at O let BD intersect RS at E and AC intersect RQ at F
IN ABD P and S are mid-points of sides AB and AD.
PS||BD and PS=
Similarly, RQ||DB and RQ=BD
RS||BD||RQ and PS =
PS=RQ and PS||RQ
PQRS is a parallelogram
Now RF||EO and RE||FO
OFRE is also a parallelogram.
Again, we know that diagonals of a rhombus bisect each other at right angles.
Each angle of the parallelogram PQRS is
Hence PQRS is a rectangle.
10. In the given Fig ABCD is a parallelogram E is mid-point of AB and CE bisects Prove that:
(i) AE = AD
(ii) DE bisects
(iii)
Ans. ABCD is a parallelogram
And EC cuts them
(i) Now AE=AD
[
Alternate interior angles]
(ii) DE bisects
(iii) Now
But, the sum of all the angles of the triangle is
11. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. show that
(i) D is mid-point of AC
(ii) MDAC
(iii) CM = MA = AB
Ans. Given ABC is a right angle at C
(i) M is mid-point of AB
And MD||BC
D is mid-Point of AC [a line through midpoint of one side of a
parallel to another side bisect the third side.
(ii). MD||BC
(iii) In ADM and
CDM
AD=DC [D is mid-point of AC]
DM=DM
ADM
CDM
AM=CM
AM=CM=MB [ M is mid-point of AB]
CM=MA=
AB.
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