Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given: `square` ABCD is a parallelogram.
AP = BQ = CR = DS
To prove: `square`PQRS is a parallelogram.
Proof:
AP = CR ...(Given) ...(i)
`square`ABCD is a parallelogram.
AB = CD ...(opposite sides of parallelogram)
∴ AP + PB = CR + RD ...[A-P-B, D-R-C] ...(ii)
∴ PB = RD ...[From (i) and (ii)] ...(iii)
∠ABC ≅ ∠ADC ...(Opposite angles of parallelogram)
That is, ∠PBQ ≅ ∠RDS ...(A-P-B, B-Q-C, C-R-D and A-S-D) ...(iv)
In ΔPBQ and ΔRDS,
Seg PB ≅ Seg RD ...[From (iii)]
∠PBQ ≅ ∠RDS ...[From (iv)]
Seg BQ ≅ Seg SD ...(Given)
∴ ΔPBQ ≅ ΔRDS ...[SAS test]
Seg PQ ≅ Seg RS ...(c.s.c.t) ...(v)
Thus, we can prove that, ΔPAS ≅ ΔRCQ,
∴ Seg PS ≅ Seg RQ ...(vi)
In `square`PQRS,
Seg PQ ≅ Seg RS ...[From (v)]
Seg PS ≅ Seg RQ ...[from (vi)]
If the opposite sides of a quadrilateral are congruent, then it is a parallelogram.
∴ `square`PQRS is a parallelogram.
APPEARS IN
संबंधित प्रश्न
Consider the given parallelogram. Find the values of the unknowns x, y, z.
Can a quadrilateral ABCD be a parallelogram if ∠D + ∠B = 180°?
Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
The following figure GUNS is a parallelogram. Find x and y. (Lengths are in cm)
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.
Construct ☐ PQRS, such that l(PQ) = 3.5 cm, l(QR) = 5.6 cm, l(RS) = 3.5 cm, m∠Q = 110°, m∠R = 70°. If it is given that ☐ PQRS is a parallelogram, which of the given information is unnecessary?
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 PQRS, ∠Q = (4x – 5)° and ∠S = (3x + 10)°. Calculate: ∠Q and ∠R.
ABCD is a parallelogram. What kind of quadrilateral is it if: AC is perpendicular to BD but is not equal to it?
The given figure shows parallelogram ABCD. Points M and N lie in diagonal BD such that DM = BN.
Prove that:
(i) ∆DMC = ∆BNA and so CM = AN
(ii) ∆AMD = ∆CNB and so AM CN
(iii) ANCM is a parallelogram.
In the given figure, AB || EC, AB = AC and AE bisects ∠DAC. Prove that:
- ∠EAC = ∠ACB
- ABCE is a 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
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 d and f
Which of the following figures satisfy the following properties?
- All sides are congruent.
- All angles are right angles.
- Opposite sides are parallel.
If two adjacent angles of a parallelogram are (5x – 5)° and (10x + 35)°, then the ratio of these angles is ______.
The adjacent sides of a parallelogram are 5 cm and 9 cm. Its perimeter is ______.
In parallelogram MODE, the bisector of ∠M and ∠O meet at Q, find the measure of ∠MQO.
Find the values of x and y in the following parallelogram.
Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
