# Measure of the Angles of the Cyclic Quadrilateral

We will prove that, in the figure ABCD is a cyclic quadrilateral and the tangent to the circle at A is the line XY. If ∠CAY : ∠CAX = 2 : 1 and AD bisects the angle CAX while AB bisects ∠CAY then find the measure of the angles of the cyclic quadrilateral. Also, prove that DB is a diameter of the circle.

Solution:

∠CAY + ∠CAX = 180° and ∠CAY : ∠CAX = 2 : 1.

Therefore, ∠CAY = $$\frac{2}{3}$$ × 180° = 120° and ∠CAX = $$\frac{1}{3}$$ × 180° = 60°.

As AD bisects ∠CAX, ∠DAX = ∠CAD = $$\frac{1}{2}$$ × 60° = 30°

As AB bisects ∠CAY, ∠YAB = ∠CAB = $$\frac{1}{2}$$ × 120° = 60°.

Now, ∠CAY = ∠ADC = 120° (Since, angle between tangent and chord is equal to the angle in the alternate segment).

Therefore, ∠CBA = 180° - ∠ADC = 180° - 120° = 60° (Since opposite angles of a cyclic quadrilateral are supplementary).

Again, ∠DAB = ∠DAC + ∠CAB = 30° + 60° = 90°.

Therefore, ∠BCD = 180° - ∠DAB = 180° - 90° = 90°.

We can see that the chord DB subtends a right angle at A.

Therefore, DB is a diameter of the circle (as an angle in a semicircle is a right-angle).

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

Apr 20, 24 05:39 PM

There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…

2. ### What are Parallel Lines in Geometry? | Two Parallel Lines | Examples

Apr 20, 24 05:29 PM

In parallel lines when two lines do not intersect each other at any point even if they are extended to infinity. What are parallel lines in geometry? Two lines which do not intersect each other

3. ### Perpendicular Lines | What are Perpendicular Lines in Geometry?|Symbol

Apr 19, 24 04:01 PM

In perpendicular lines when two intersecting lines a and b are said to be perpendicular to each other if one of the angles formed by them is a right angle. In other words, Set Square Set Square If two…

4. ### Fundamental Geometrical Concepts | Point | Line | Properties of Lines

Apr 19, 24 01:50 PM

The fundamental geometrical concepts depend on three basic concepts — point, line and plane. The terms cannot be precisely defined. However, the meanings of these terms are explained through examples.