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Quadrilaterals — Maths STD 9 — Question

CBSE BoardEnglish MediumSTD 9MathsQuadrilaterals4 Marks
Question
ABCD is a rhombus. Show that diagonal AC bisects $\angle\text{A}$ as well as $\angle\text{C}$ and diagonal BD bisects $\angle\text{B}$ as well as $\angle\text{D}.$ 
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Answer



Let us join AC. In $\triangle\text{ABC,}$ BC = AB (Sides of a rhombus are equal to each other) $\therefore \angle1=\angle2$ (Angles opposite to equal sides of a triangle are equal) However, $ \angle1=\angle3$ (Alternate interior angles for parallel lines AB and CD) $ \Rightarrow \angle2=\angle3$ Therefore, $ \Rightarrow \angle2=\angle4$ AC bisects $\angle\text{C}.$ Also, (Alternate interior angles for || lines BC and DA) $ \Rightarrow \angle1=\angle4$ Therefore, AC bisects $\angle\text{A}.$ Similarly, it can be proved that BD bisects $\angle\text{B}$ and $\angle\text{D}$ as well.

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