Question
In the circle shown here, chords AC¯¯¯¯¯¯¯¯ and BD¯¯¯¯¯¯¯¯ intersect at point E , which is not the center of the circle. Find m∠BEC if BC measures 60° and AD measures 40°.
Answers
GPT-4o mini
To find the measure of angle \( m\angle BEC \) formed by the intersecting chords \( AC \) and \( BD \) at point \( E \), we can use the angle formed by two chords theorem, which states that:
\[
m\angle BEC = \frac{1}{2} (m\angle BAE + m\angle DCE)
\]
In this case, we know:
- \( m\angle BAE = 40^\circ \)
- \( m\angle DCE = 60^\circ \)
Now we can substitute these values into the formula:
\[
m\angle BEC = \frac{1}{2} (40^\circ + 60^\circ)
\]
Calculating inside the parentheses first:
\[
40^\circ + 60^\circ = 100^\circ
\]
Then, divide by 2:
\[
m\angle BEC = \frac{1}{2} \times 100^\circ = 50^\circ
\]
Thus, the measure of angle \( m\angle BEC \) is:
\[
\boxed{50^\circ}
\]
\[
m\angle BEC = \frac{1}{2} (m\angle BAE + m\angle DCE)
\]
In this case, we know:
- \( m\angle BAE = 40^\circ \)
- \( m\angle DCE = 60^\circ \)
Now we can substitute these values into the formula:
\[
m\angle BEC = \frac{1}{2} (40^\circ + 60^\circ)
\]
Calculating inside the parentheses first:
\[
40^\circ + 60^\circ = 100^\circ
\]
Then, divide by 2:
\[
m\angle BEC = \frac{1}{2} \times 100^\circ = 50^\circ
\]
Thus, the measure of angle \( m\angle BEC \) is:
\[
\boxed{50^\circ}
\]