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ABCD is a rhombus. Show that diagonal AC bisects ∠ A as well as ∠ C and diagonal BD bisects ∠ B as well as ∠ D.

Q: 7  ABCD is a rhombus. Show that diagonal AC bisects \small \angle A as well as \small \angle C and diagonal BD bisects \small \angle B as well as \small \angle D.
 

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In \triangle ADC,

   AD = CD          (ABCD is a rhombus)

 \angle3=\angle1.................1(angles opposite to equal sides are equal )

 \angle3=\angle2.................2 (alternate angles)

From 1 and 2, we have 

                       \angle1=\angle2.................3

   and        \angle1=\angle4.................4 (alternate angles)

From 1 and 4, we get 

                        \angle3=\angle4.................5

Hence, from 3 and 5, diagonal AC bisects angles A as well as angle C.

In \triangle ADB,

   AD = AB          (ABCD is a rhombus)

 \angle5=\angle7.................6(angles opposite to equal sides are equal )

 \angle7=\angle6.................7 (alternate angles)

From 6 and 7, we have 

                       \angle5=\angle6.................8

   and        \angle5=\angle8.................9(alternate angles)

From 6 and 9, we get 

                        \angle7=\angle8.................10

Hence, from 8 and 10, diagonal BD bisects angles B as well as angle D.

           

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