Question

Abcd is a cyclic quadrilateral ac is a diameter of the circle mn is the tangent to d cad angle is 40 and acb is 55 find the measure of angle adm and bad

Answer 1

Use the properties of tangent to solve this type is  sums!!

Answer 2

The measure of ∠ ADM = 50° and ∠ BAD = 75°.

Step-by-step explanation:

Given:  

Here, ABCD is a cyclic quadrilateral.

AC is a diameter of the circle and MN is the tangent at D.

Also, ∠ CAD = 40, ∠ ACB = 55.

As shown in the figure, let ∠ADM = x, ∠ACD = y, ∠BAC = z

To find:

The measure of ∠ ADM and ∠ BAD

Solution:

Here angle in the alternate segments are equal,

so ∠ CDN = ∠ DAC = 40°

Now since AC is diameter,

so∠ADC = ∠ABC = 90°  (Angle in a semi circle is right angle)

Hence using linear pair property,

x + 90° + 40° = 180°

x + 130° = 180°

x = 180° - 130°

x = 50°

∠ ADM = x = 50°.                [ shown in figure ]

Now,  y = x = 50°  [Angles in alternate segment are equal]

and opposite angles of cyclic quadrilateral are supplementary, then

∠ A+ ∠C = 180°

z + 40° + 55°  +50° = 180°

z + 145° = 180°

z  = 180° - 145°

z = 35°

So ∠ BAD = z + 40 = 35 + 40 = 75°