Explore the properties of angles formed by two intersecting chords.

✓ 1.The intersecting chords form vertical angles. If m∠DEB = 105°,
then m∠AEC = °.

✓ 2. Find the sum of the angle measures:
m∠DEB + m∠AEC = °

✓ 3. Find the sum of the arc measures:
#& m\overset{\frown}{\sf BD} + m\overset{\frown}{\sf AC} = #& °

✓ 4. The sum of the angle measures was equal to the sum of the arc measures. Move points B and C to change the angles and arcs.

Make a conjecture. Is the sum of the angle measures always equal to the sum of the arc measures? yes no

✓ 5. Because vertical angles are congruent, m∠DEB = m∠AEC. Therefore,
m∠DEB + m∠AEC =
1/2 2 × m∠DEB.

✓ 6. By addition substitution #&{ \sf 2}(m \angle {\sf DEB}) = m\overset{\frown}{\sf BD} + m\overset{\frown}{\sf AC},#& or

#&m \angle {\sf DEB} = { \large \frac{\sf 1}{\sf 2}} (m\overset{\frown}{\sf BD} + m\overset{\frown}{\sf AC}).#&