To find the measure of angle \( m\angle FDE \), we will use the property of angles formed by chords in a circle. Specifically, the measure of the angle formed by two chords meeting at a point inside the circle is equal to half the sum of the measures of the arcs intercepted by the angle.
In this case, arcs \( BC \) and \( EF \) intercepted by angle \( EDF \) are given as follows:
- \( m\text{arc } BC = 67^\circ \)
- \( m\text{arc } EF = 74^\circ \)
To find \( m\angle FDE \), we first need to find the measure of angle \( EDF \). The relationship is given by:
\[ m\angle EDF = \frac{1}{2} (m\text{arc } BC + m\text{arc } EF) \]
Substituting the values of the arcs:
\[ m\angle EDF = \frac{1}{2} (67^\circ + 74^\circ) = \frac{1}{2} (141^\circ) = 70.5^\circ \]
The angle \( FDE \) is inscribed in the same segment as \( EDF \) and has the same measure as \( EDF \).
Thus,
\[ m\angle FDE = m\angle EDF = 70.5^\circ \]
Therefore, the answer is \( \boxed{70.5^\circ} \).
1 answer