(i) In an isosceles triangle the angles are in the ratio 7 : 4 : 7. Find each base angle of the triangle.

(ii) Find the angles of an isosceles triangle, if the ratio of the base angle to the vertical angle 2 : 5.

(i) Given: The angles of the isosceles triangle are in the ratio 7 : 4 : 7.

Let the three angles be 7a, 4a, and 7a.

Using the angle sum property of a triangle:

⇒ 7a + 4a + 7a = 180°

⇒ 18a = 180°

⇒ a = $\dfrac{180°}{18}$

⇒ a = 10°

Now, the two base angles are:

7a = 7 x 10° = 70°

Hence, the base angles are 70° and 70°.

(ii) Given: The ratio of the base angle to the vertical angle is 2 : 5.

Let each base angle be 2a and the vertical angle be 5a.

Using the angle sum property of a triangle:

⇒ 2a + 5a + 2a = 180°

⇒ 9a = 180°

⇒ a = $\dfrac{180°}{9}$

⇒ a = 20°

Now, the angles of the triangle are:

2a = 2 x 20° = 40° and 5a = 5 x 20° = 100°

Hence, the angles of isosceles triangle are 40°, 40° and 100°.