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Quadrilaterals — Maths STD 9 — Question

Maharashtra BoardEnglish MediumSTD 9MathsQuadrilaterals5 Marks
Question
ABCD is a rectangle in which diagonal AC bisects $\angle\text{A}$ as well as $\angle\text{C}.$ Show that:
  1. ABCD is a square
  2. diagonal BD bisects $\angle\text{B}$ as well as $\angle\text{D}.$​​​​​​​
✓

Answer


  1. ABCD is a rectangle in which diagonal AC bisects $\angle\text{A}$ as well as $\angle\text{C}.$
$\Rightarrow\angle\text{BAC}=\angle\text{DAC}...(1)$

And $\angle\text{BCA}=\angle\text{DCA}...(2)$

Since every rectangle is a parallelogram, therefore

AB || DC and AC is the transversal.

$\Rightarrow\angle\text{BAC}=\angle\text{DCA}$ (alternate angles)

$\Rightarrow\angle\text{DAC}=\angle\text{DCA}$ [from]

Thus in $\triangle\text{ADC},$

$\text{AD = CD}$ (opposite sides of equal angles are equal)

But, $\text{AD = BC}$ and $\text{CD = AB}$ (ABCD is a rectangle)

$\Rightarrow\text{AB = BC = CD = AD}$

Hence, ABCD is a square.
  1. In $\triangle\text{BAD}$ and $\triangle\text{BCD},$
$\text{AB = CD}$

$\text{AD = BC}$

$\text{BD = BD}$

$\therefore\triangle\text{BAD}\cong\triangle\text{BCD}$ (by SSS congruence criterion)

$\Rightarrow\angle\text{ABD}=\angle\text{CBD}$ and $\angle\text{ADB}=\angle\text{CDB}$ [CP.C.T.]

Hence, diagonal BD bisects $\angle\text{B}$ as well as $\angle\text{D}.$

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