Rajan is studying in IX standard. His father purchased a plot which is in a square shape. Aftervisiting the land, few questions came in his mind. Give answers to his questions by looking at thefigure.Answer the following:i) Find the measure of ∠AOB.ii) Which is the correct congruence rule applicable to prove ∆ABO ≅ ∆ADO?iii) If OB = 5cm, then find the value of BD.iv) Which is the correct congruence rule applicable to prove ∆ABC ≅ ∆BAD?

i) To find the measure of ∠AOB, we can use the fact that the sum of angles in a square is 360 degrees. Since the square has four equal angles, each angle measures 360/4 = 90 degrees. Therefore, ∠AOB measures 90 degrees.

ii) To prove ∆ABO ≅ ∆ADO, we can use the Side-Angle-Side (SAS) congruence rule. This rule states that if two sides and the included angle of one triangle are congruent to the corresponding sides and included angle of another triangle, then the triangles are congruent. In this case, we can see that side AB is congruent to side AD, side AO is congruent to side AO (common side), and angle AOB is congruent to angle AOD (both are right angles). Therefore, we can use the SAS congruence rule to prove ∆ABO ≅ ∆ADO.

iii) If OB = 5cm, we can find the value of BD by using the Pythagorean theorem. In triangle OBD, OB is the hypotenuse and BD is one of the legs. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, we have OB^2 = BD^2 + OD^2. Substituting the given values, we get 5^2 = BD^2 + 5^2. Simplifying this equation, we have 25 = BD^2 + 25. By subtracting 25 from both sides, we get BD^2 = 0. Taking the square root of both sides, we get BD = 0. Therefore, the value of BD is 0 cm.

iv) To prove ∆ABC ≅ ∆BAD, we can use the Side-Side-Side (SSS) congruence rule. This rule states that if the three sides of one triangle are congruent to the corresponding sides of another triangle, then the triangles are congruent. In this case, we can see that side AB is congruent to side BA (common side), side BC is congruent to side AD (both are equal to 5 cm), and side AC is congruent to side BD (both are equal to 0 cm). Therefore, we can use the SSS congruence rule to prove ∆ABC ≅ ∆BAD.