Question

23. A, B, C and D are points on the circumference of a circle, centre O. 258 AC is a diameter of the circle. AC and BD intersect at E. Angle CAB= 25° Angle DEC= 100° Work out the size of angle DAC. You must show all your working. ​

Answer 1

Answer:

Given that \(AC\) is a diameter of the circle, it means that angle \(ACB\) is a right angle, which is 90°.

Now, since \(AC\) and \(BD\) intersect at \(E\), and \(AC\) is a diameter, it means \(E\) is the midpoint of \(AC\). Therefore, angle \(DEC\) is subtended by the same arc \(AC\) as angle \(CAB\).

Using the property that angles subtended by the same arc are equal, we have:

\[ \text{Angle } DEC = \text{Angle } CAB \]

So, angle \(DEC\) is also 25°.

Now, we need to find angle \(DAC\). We know that angle \(DEC\) is 100° and angle \(DEC\) is made up of angles \(DEA\) and \(EAC\). Angle \(EAC\) is equal to angle \(CAB\) which is 25°.

Therefore, angle \(DEA = 100° - 25° = 75°\).

Now, we have angle \(DEA\) and angle \(DAC\) as vertical angles, and vertical angles are equal.

Thus, angle \(DAC = 75°\).