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
Given a parallelogram ABCD. Complete each statement along with the definition or property used.
- AD = ______
- ∠DCB = ______
- OC = ______
- m∠DAB + m∠CDA = ______
Consider the given parallelograms. Find the values of the unknowns x, y, z.
Consider the given parallelograms. Find the values of the unknowns x, y, z.
Can a quadrilateral ABCD be a parallelogram if ∠D + ∠B = 180°?
The following figure GUNS is a parallelogram. Find x and y. (Lengths are in cm)
In the above figure both RISK and CLUE are parallelograms. Find the value of x.
In the given figure, `square`PQRS and `square`ABCR are two parallelograms. If ∠P = 110° then find the measures of all angles of `square`ABCR.
In the given figure, G is the point of concurrence of medians of ΔDEF. Take point H on ray DG such that D-G-H and DG = GH, then prove that `square`GEHF is a parallelogram.
Ratio of consecutive angles of a quadrilateral is 1 : 2 : 3 : 4. Find the measure of its each angle. Write, with reason, what type of a quadrilateral it is.
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 ABCD, E is the mid-point of AD and F is the mid-point of BC. Prove that BFDE is a parallelogram.
In parallelogram ABCD, E is the mid-point of side AB and CE bisects angle BCD. Prove that:
- AE = AD,
- DE bisects and ∠ADC and
- Angle DEC is a right angle.
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 d and e
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
In the given figure, ABCD and BDCE are parallelograms with common base DC. If BC ⊥ BD, then ∠BEC = ______.
The adjacent sides of a parallelogram are 5 cm and 9 cm. Its perimeter is ______.
In the following figure, FD || BC || AE and AC || ED. Find the value of x.
Construct a parallelogram when one of its side is 4 cm and its two diagonals are 5.6 cm and 7 cm. Measure the other side.
