Question

In cyclic quadrilateral ABCD m\(\angle\)A = 100\(^\circ\), then find m\(\angle\)C.

Hint:

Always remember the main property of cyclic quadrilaterals: opposite angles add up to 180\(^\circ\). This is a fundamental theorem frequently tested in geometry problems.

Answer 1

Step 1: Understanding the Concept:
A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. A key property of cyclic quadrilaterals is related to its opposite angles.

Step 2: Key Formula or Approach:
The theorem of cyclic quadrilateral states that the sum of opposite angles of a cyclic quadrilateral is 180\(^\circ\) (supplementary). \[ m\angle A + m\angle C = 180^\circ \] \[ m\angle B + m\angle D = 180^\circ \]

Step 3: Detailed Explanation:
Given: \[\begin{array}{rl} \bullet & \text{ABCD is a cyclic quadrilateral.} \\ \bullet & \text{\(m\angle A = 100^\circ\).} \\ \end{array}\] Since ABCD is a cyclic quadrilateral, its opposite angles are supplementary. Therefore, the sum of \(\angle A\) and its opposite angle \(\angle C\) is 180\(^\circ\). \[ m\angle A + m\angle C = 180^\circ \] Substitute the given value of \(m\angle A\): \[ 100^\circ + m\angle C = 180^\circ \] \[ m\angle C = 180^\circ - 100^\circ \] \[ m\angle C = 80^\circ \]

Step 4: Final Answer:
The measure of \(\angle C\) is 80\(^\circ\).