Advertisements
Advertisements
Question
In a parallelogram ABCD, E is the midpoint of AB and DE bisects angle D. Prove that:CE is the bisector of angle C and angle DEC is a right angle
Advertisements
Solution
Since BC = BE
⇒ ∠BEC = ∠BCE ...(Angles opposite to equal sides are equal)
∠BEC = ∠ECD ...(Alternate angles)
⇒ ∠BCE = ∠ECD
⇒ CE is the bisector of ∠C ....(proved)
∠DCE = `(1)/(2)∠"C"` ...(Given CE bisects ∠D)
∠CDE = `(1)/(2)∠"D"` ...(Given DE bisects ∠D)
∠DCE + ∠CDE
= `(1)/(2)(∠"C" + ∠"D")`
= `(1)/(2) xx 180°` = 90°
Thus, in ΔDCE,
∠DEC = 180° - ∠DCE + ∠CDE = 180° - 90°
⇒ ∠DEC = 90°.
APPEARS IN
RELATED QUESTIONS
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 following figure shows a trapezium ABCD in which AB is parallel to DC and AD = BC. 
Prove that:
(i) ∠DAB = ∠CBA
(ii) ∠ADC = ∠BCD
(iii) AC = BD
(iv) OA = OB and OC = OD.
In parallelogram ABCD, the bisector of angle A meets DC at P and AB = 2 AD.
Prove that:
(i) BP bisects angle B.
(ii) Angle APB = 90o.
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 a parallelogram ABCD, E is the midpoint of AB and DE bisects angle D. Prove that: BC = BE.
In the given figure, the perimeter of parallelogram PQRS is 42 cm. Find the lengths of PQ and PS.
In the Figure, ABCD is a rectangle and EFGH is a parallelogram. Using the measurements given in the figure, what is the length d of the segment that is perpendicular to `bar("HE")` and `bar("FG")`?
Which of the following statement is correct?
In the following figure, it is given that BDEF and FDCE are parallelograms. Can you say that BD = CD? Why or why not?
Construct a parallelogram POUR in which, PO = 5.5 cm, OU = 7.2 cm and ∠O = 70°.
