Q. 84.1( 435 Votes )
ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C Show that:
(i) ABCD is a square
(ii) Diagonal BD bisects ∠B as well as ∠D
Answer :
Given: ABCD is a rectangle,
∠A = ∠C
∠A = 
∠C
To Prove: ABCD is a square
Proof:
∠DAC =∠ DCA (AC bisects A and C)
CD = DA (Sides opposite to equal angles are also equal)
However,
DA = BC and AB = CD (Opposite sides of a rectangle are equal)
AB = BC = CD = DA
ABCD is a rectangle and all of its sides are equal.
Hence, ABCD is a square
Hence, Proved.(ii) Let us join BD
In ΔBCD,
BC = CD (Sides of a square are equal to each other)
∠CDB = ∠CBD (Angles opposite to equal sides are equal)
However,
∠CDB = ∠ABD (Alternate interior angles for AB || CD)
∠CBD = ∠ABD
BD bisects B
Also,
∠CBD = ∠ADB (Alternate interior angles for BC || AD)
∠CDB = ∠ABD
BD bisects D
Rate this question :
Critical Thinking Problems on Quadrilaterals44 mins
Quiz | Properties of Parallelogram31 mins
Extras on Quadrilaterals40 mins
Smart Revision | Quadrilaterals43 mins
Quiz | Basics of Quadrilaterals36 mins
RD Sharma | Extra Qs. of Cyclic Quadrilaterals31 mins
If the diagonals of a rhombus are 18 cm and 24 cm respectively, then its side is equal to
RD Sharma - MathematicsIn a parallelogram ABCD, if ∠DAB=75° and ∠DBC=60°, then ∠BDC=
RD Sharma - MathematicsThe diagonals AC and BD of a rectangle ABCD intersect each other at P. If ∠ABD =50°, then ∠DPC =
RD Sharma - MathematicsIn a quadrilateral ABCD, ∠A+∠C is 2 times ∠B+∠D. If ∠A= 140° and f∠D = 60°, then ∠B=
RD Sharma - MathematicsIf one side of a parallelogram is 24° less than twice the smallest angle, then the measure of the largest angle of the parallelogram is
RD Sharma - MathematicsIf the degree measures of the angles of quadrilateral are 4x, 7x, 9x and 10x, what is the sum of the measures of the smallest angle and largest angle?
RD Sharma - MathematicsIf an angle of a parallelogram is two-third of its adjacent angle, the smallest angle of the parallelogram is
RD Sharma - MathematicsIn Fig. 14.112, ABCD is a trapezium. Find the values of x and y.
