Advertisements
Advertisements
प्रश्न
In the adjacent figure, if seg AB || seg PQ, seg AB ≅ seg PQ, seg AC || seg PR, seg AC ≅ seg PR then prove that, seg BC || seg QR and seg BC ≅ seg QR.
Advertisements
उत्तर
Given: seg AB || seg PQ, seg AB ≅ seg PQ,
seg AC || seg PR, seg AC ≅ seg PR
To prove: seg BC || seg QR, seg BC ≅ seg QR
Proof:
In `square`ABQP,
seg AB || seg PQ
seg AB ≅ seg PQ ...(Given)
∴ `square`ABQP is a parallelogram. ...(A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and congruent)
∴ segAP || segBQ ...(i)
∴ seg AP ≅ seg BQ ...(ii) ...(Opposite sides of a parallelogram)
In `square`ACRP,
seg AC || seg PR
seg AC ≅ seg PR ...(Given)
∴ `square`ACRP is a parallelogram. ...(A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and congruent)
∴ seg AP || seg CR ...(iii)
∴ seg AP ≅ seg CR ...(iv) ...(Opposite sides of a parallelogram)
In `square`BQRC,
seg BQ || seg CR ...[From (i) and (iii)]
seg BQ ≅ seg CR ...[From (ii) and (iv)]
∴ `square`BQRC is a parallelogram. ...(A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and congruent)
∴ seg BC || seg QR
∴ seg BC ≅ seg QR ...(Opposite sides of parallelogram)
APPEARS IN
संबंधित प्रश्न
Can a quadrilateral ABCD be a parallelogram if ∠D + ∠B = 180°?
Can a quadrilateral ABCD be a parallelogram if AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
Can a quadrilateral ABCD be a parallelogram if ∠A = 70° and ∠C = 65°?
In the given figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that `square`PQRS is a parallelogram.
Construct ☐ BARC such that l(BA) = l(BC) = 4.2 cm, l(AC) = 6.0 cm, l(AR) = l(CR) = 5.6 cm
Referring the adjacent figure of a parallelogram, write the answer of questions given below.
(1) If l(WZ) = 4.5 cm then l(XY) = ?
(2) If l(YZ) = 8.2 cm then l(XW) = ?
(3) If l(OX) = 2.5 cm then l(OZ) = ?
(4) If l(WO) = 3.3 cm then l(WY) = ?
(5) If m∠WZY = 120° then m∠WXY = ? and m∠XWZ = ?
In parallelogram ABCD, ∠A = 3 times ∠B. Find all the angles of the parallelogram. In the same parallelogram, if AB = 5x – 7 and CD = 3x +1 ; find the length of CD.
In parallelogram PQRS, ∠Q = (4x – 5)° and ∠S = (3x + 10)°. Calculate: ∠Q and ∠R.
Prove that the diagonals of a parallelogram bisect each other.
In parallelogram ABCD, E is the mid-point of AD and F is the mid-point of BC. Prove that BFDE is a parallelogram.
In the given figure, AB || EC, AB = AC and AE bisects ∠DAC. Prove that:
- ∠EAC = ∠ACB
- ABCE is a parallelogram.
If a triangle and a parallelogram lie on the same base and between the same parallels, then prove that the area of the triangle is equal to half of the area of parallelogram
Iron rods a, b, c, d, e, and f are making a design in a bridge as shown in the figure. If a || b, c || d, e || f, find the marked angles between b and c
In parallelogram LOST, SN ⊥ OL and SM ⊥ LT. Find ∠STM, ∠SON and ∠NSM.
ABCD is a parallelogram. Find the value of x, y and z.
A diagonal of a parallelogram bisects an angle. Will it also bisect the other angle? Give reason.
In parallelogram ABCD, the angle bisector of ∠A bisects BC. Will angle bisector of B also bisect AD? Give reason.
Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
