Advertisements
Advertisements
Question
In the given figure, `square`ABCD is a parallelogram. Point E is on the ray AB such that BE = AB then prove that line ED bisects seg BC at point F.
Advertisements
Solution
`square`ABCD is a parallelogram. ...[Given]
∴ seg AB ≅ seg DC ...(i) [Opposite angles of a parallelogram]
seg AB ≅ seg BE ...(ii) [Given]
seg DC ≅ seg BE ...(iii) [From (i) and (ii)]
side DC || side AB ...[Opposite sides of a parallelogram]
i.e. side DC || seg AE and seg DE is their transversal. ...[A-B-E]
∴ ∠CDE ≅ ∠AED
∴ ∠CDF ≅ ∠BEF ...(iv) [D-F-E, A-B-E]
In ∆DFC and ∆EFB,
seg DC = seg EB ...[From (iii)]
∠CDF ≅ ∠BEF ...[From (iv)]
∠DFC ≅ ∠EFB ...[Vertically opposite angles]
∴ ∆DFC ≅ ∆EFB ...[SAA test]
∴ FC ≅ FB ...[c.s.c.t]
∴ Line ED bisects seg BC at point F.
APPEARS IN
RELATED QUESTIONS
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°?
The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.
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.
If the ratio of measures of two adjacent angles of a parallelogram is 1 : 2, find the measures of all angles of the parallelogram.
Construct a parallelogram ABCD such that l(BC) = 7 cm, m∠ABC = 40° , l(AB) = 3 cm.
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.
PQRS is a parallelogram whose diagonals intersect at M.
If ∠PMS = 54°, ∠QSR = 25° and ∠SQR = 30° ; find :
(i) ∠RPS
(ii) ∠PRS
(iii) ∠PSR.
Given: Parallelogram ABCD in which diagonals AC and BD intersect at M.
Prove: M is the mid-point of LN.
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 following diagram, the bisectors of interior angles of the parallelogram PQRS enclose a quadrilateral ABCD.
Show that:
(i) ∠PSB + ∠SPB = 90°
(ii) ∠PBS = 90°
(iii) ∠ABC = 90°
(iv) ∠ADC = 90°
(v) ∠A = 90°
(vi) ABCD is a rectangle
Thus, the bisectors of the angles of a parallelogram enclose a rectangle.
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 c and f
Which of the following is a property of a parallelogram?
The angle between the two altitudes of a parallelogram through the same vertex of an obtuse angle of the parallelogram is 30°. The measure of the obtuse angle is ______.
If opposite angles of a quadrilateral are equal, it must be a parallelogram.
A diagonal of a parallelogram bisects an angle. Will it also bisect the other angle? 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.
ABCD is a parallelogram. Points P and Q are taken on the sides AB and AD respectively and the parallelogram PRQA is formed. If ∠C = 45°, find ∠R.
Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
