Answer

Verified

385.5k+ views

**Hint:**To solve this question we need to know the basic theory related to the rectangle. We learned that a rectangle has four-sided and all the internal angles are equal to 90 degrees. The length of the opposite sides of the rectangle is always equal to each other. The length of the opposite sides of the rectangle is always equal to each other. If all the sides are equal then it is square.

**Complete step-by-step solution:**

(i) we need to prove ABCD is square.

Given in question,

ABCD is a rectangle, in which diagonal AC bisect $\angle $${\text{A}}$ as well as $\angle $${\text{C}}$.

Therefore,

$\angle $${\text{DAC}}$=$\angle $${\text{CAB}}$……………… (1)

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

A square is a rectangle when all sides are equal. Now,

AD$\parallel $BC & AC is transversal, therefore

$\angle$${\text{DAC}}$=$\angle$${\text{BCA}}$ [Alternate angles]

From (1),

$\angle $${\text{CAB}}$=$\angle $${\text{BCA}}$……………….. (3)

In $\vartriangle $ABC,

$\angle $${\text{CAB}}$=$\angle $${\text{BCA}}$, therefore

BC=AB………………….. (4) [sides opposite to equal angles]

But BC=AD & AB=DC………………….. (5) [Opposite sides of rectangle]

Therefore from (4) & (5),

AB=BC=CD=AD

**Hence, ABCD is a square.**

(ii) we need to prove that diagonal BD bisects $\angle $${\text{B}}$ as well as $\angle $${\text{D}}$.

Now, As we discussed in the previous part.

ABCD is a square and we know that diagonals of a square bisect its angles.

**Hence, BD bisects ∠B as well as ∠D.**

**Note:**Here we use various properties of a rectangle. Always remember that the diagonals of a rectangle bisect each other and also the sides in a rectangle are always equal to each other.

Recently Updated Pages

How do you evaluate cos left dfrac13pi 12 right class 10 maths CBSE

How do you rewrite the inequality left 11 2x right class 10 maths CBSE

How do you solve 4 3x 025 class 10 maths CBSE

How do you find the zeros of x3 3x2 + 6x 18 class 10 maths CBSE

Consider the following statements in respect of the class 10 maths CBSE

How do you factor 2x3 + 3x2 8x 12 class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail

Name 10 Living and Non living things class 9 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths