Advertisements
Advertisements
प्रश्न
The diagonal BD of a parallelogram ABCD bisects angles B and D. Prove that ABCD is a rhombus.
Advertisements
उत्तर
Given: ABCD is a parallelogram where the diagonal BD bisects
parallelogram ABCD at angle B and D
To Prove: ABCD is a rhombus
Proof: Let us draw a parallelogram ABCD where the diagonal BD bisects the parallelogram at an angle B and D.
Construction: Let us join AC as a diagonal of the parallelogram ABCD
Since ABCD is a parallelogram in which diagonal BD bisects ∠B and ∠D
BD bisect ∠B,
∠ABD = ∠CBD = `1/2`∠ABC
BD bisect ∠D,
∠ADB = ∠CDB = `1/2`∠ABC
∴ ∠ABC = ∠ADC ...(Opposite angles of parallelogram are equal)
∴ `1/2`∠ABC = `1/2` ∠ADC
∠ABD = ∠ADB and ∠CBD = ∠CDB
∴ AD = AB and CD = BC
As ABCD is a parallelogram, opposite sides are equal
AB = CD and AD = BC
AB = BC = CD = AD
∴ ABCD is a rhombus.
APPEARS IN
संबंधित प्रश्न
E is the mid-point of side AB and F is the mid-point of side DC of parallelogram ABCD. Prove that AEFD is a parallelogram.
The alongside figure shows a parallelogram ABCD in which AE = EF = FC.
Prove that:
- DE is parallel to FB
- DE = FB
- DEBF is a parallelogram.
In the given figure, ABCD is a parallelogram.
Prove that: AB = 2 BC.
PQRS is a parallelogram. T is the mid-point of PQ and ST bisects ∠PSR.
Prove that: RT bisects angle R
PQRS is a parallelogram. T is the mid-point of PQ and ST bisects ∠PSR.
Prove that: ∠RTS = 90°
ABCD is a parallelogram. The bisector of ∠BAD meets DC at P, and AD is half of AB.
Prove that: ∠APB is a right angle.
In the given figure, the perimeter of parallelogram PQRS is 42 cm. Find the lengths of PQ and PS.
Which of the following statement is correct?
In the following figure, ABCD and AEFG are two parallelograms. If ∠C = 55º, determine ∠F.
Construct a parallelogram POUR in which, PO = 5.5 cm, OU = 7.2 cm and ∠O = 70°.
