Advertisements
Advertisements
Question
In parallelogram ABCD, the angle bisector of ∠A bisects BC. Will angle bisector of B also bisect AD? Give reason.
Advertisements
Solution
Given, ABCD is a parallelogram, bisector of ∠A, bisects BC at F, i.e. ∠1 = ∠2, CF = FB.
Draw FE || BA.
ABFE is a parallelogram by construction ...[∵ FE || BA]
⇒ ∠1 = ∠6 ...[Alternate angle]
But ∠1 = ∠2 ...[Given]
∴ ∠2 = ∠6
AB = FB [Opposite sides to equal angles are equal] ...(i)
∴ ABFE is a rhombus.
Now, In ΔABO and ΔBOF,
AB = BF ...[From equation (i)]
BO = BO ...[Common]
AO = FO ...[Diagonals of rhombus bisect each other]
∴ ΔABO ≅ ΔBOF ...[By SSS]
∠3 = ∠4 ...[By CPCT]
Now, BF = `1/2` BC ...[Given]
⇒ BF = `1/2` AD ...[BC = AD]
⇒ AE = `1/2` AD ...[BF = AE]
∴ E is the midpoint of AD.
APPEARS IN
RELATED QUESTIONS
In the above figure both RISK and CLUE are parallelograms. Find the value of x.
If the ratio of measures of two adjacent angles of a parallelogram is 1 : 2, find the measures of all angles of the parallelogram.
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.
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?
In parallelogram ABCD, X and Y are midpoints of opposite sides AB and DC respectively. Prove that:
(i) AX = YC
(ii) AX is parallel to YC
(iii) AXCY 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
If opposite angles of a quadrilateral are equal, it must be a parallelogram.
Find the values of x and y in the following parallelogram.
ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. Is AXCY a parallelogram? Give reason.
The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°. Find the angles of the parallelogram.
