Diagonal AC of a parallelogram ABCD bisects ∠A (see Fig. 8.19). Show that
i) it bisects ∠C also, ii) ABCD is a rhombus

Solution:

Given: The diagonal AC of a parallelogram ABCD bisects ∠A.

We can use alternate interior angles property to show that the diagonal AC bisects ∠C and by showing all sides are equal, it can be proved ABCD is a rhombus.

i) ABCD is a parallelogram.

∠DAC = ∠BCA (Alternate interior angles)   ....................(1)

∠BAC = ∠DCA (Alternate interior angles)   ....................(2)

However, it is given that AC bisects ∠A.

∠DAC = ∠BAC                                            ....................(3)

From equations (1), (2), and (3), we obtain

∠DCA  = ∠BAC = ∠DAC = ∠BCA                ....................(4)

Thus, ∠DCA = ∠BCA

Hence, AC bisects ∠C.

ii) From Equation (4), we obtain

∠DAC = ∠DCA

DA = DC (Side opposite to equal angles are equal)

However, DA = BC and AB = CD (Opposite sides of a parallelogram  are equal)

Thus, AB = BC = CD = DA

Hence, ABCD is a rhombus.

☛ Check: NCERT Solutions for Class 9 Maths Chapter 8

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Video Solution:

Diagonal AC of a parallelogram ABCD bisects ∠A (see Fig. 8.19). Show that i) It bisects ∠C also, ii) ABCD is a rhombus

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 6

Summary:

If diagonal AC of a parallelogram ABCD bisects ∠A, then it bisects ∠C, and also it is proved that ABCD is a rhombus.

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☛ Related Questions:

• If the diagonals of a parallelogram are equal, then show that it is a rectangle.
• Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
• Show that the diagonals of a square are equal and bisect each other at right angles.
• Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.