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Quadrilaterals — Maths STD 9 — Question

CBSE BoardEnglish MediumSTD 9MathsQuadrilaterals4 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



i. $\angle\text{DAC}=\angle\text{DCA}$ (AC bisects as well as)
⇒ AD = CD (Sides opposite to equal angles of a triangle are equal) also, CD = AB (Opposite sides of a rectangle)
Therefore, AB = BC = CD = AD
Thus, ABCD is a square.
ii. In $\triangle\text{BCD}$
BC = CD
Angles opposite to equal sides are equal.
also, $ \angle\text{CDB}=\angle\text{ABD}$ (Alternate interior angles)
Thus, BD bisects. Now,
$ \angle\text{CDB}=\angle\text{ABD}$
$ \angle\text{CDB}=\angle\text{ADB}$
Thus, BD bisects $\angle\text{D}.$ 

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