Let’s work through the problems provided in the image step-by-step.

Questions 18–20

18. The sides of an inscribed angle intercept the circle at the endpoints of the diameter.
Answer: 90°.
Reason: An inscribed angle subtending the diameter always measures 90° because it forms a right angle.

19. Quadrilateral $ABCD$ is inscribed in $\bigodot O$. If $m\angle A = 70°$, find $m\angle C$.
Answer: 110°.
Reason: Opposite angles of an inscribed quadrilateral sum to 180°. Thus, $m\angle C = 180° - 70° = 110°$.

20. If inscribed angles $ABC$ and $ADC$ both intercept $\overarc{AC}$, which measures 120°, find $m\angle ABC$ and $m\angle ADC$.
Answer: $m\angle ABC = 60°$, $m\angle ADC = 60°$.
Reason: An inscribed angle is half the measure of its intercepted arc. Therefore, $m\angle ABC = m\angle ADC = \frac{1}{2}(120°) = 60°$.

Proof of $m\angle B = \frac{1}{2}m\angle AOC$

Let’s fill out the missing steps in the proof table:

1. $\overline{BC}$ is a diameter of $\bigodot O$; $\overline{AB}$ is any other chord.
   Reason: Given.

2. Draw radius $\overline{OA}$.
   Reason: Definition of radius.

3. $m\angle A + m\angle B = m\angle AOC$.
   Reason: The angles add up because they share the arc $\overarc{AC}$.

4. All radii of $\bigodot O$ are congruent.
   Reason: Radii of a circle are equal (Postulate 21).

5. $\triangle OAB$ is isosceles.
   Reason: Isosceles Triangle Theorem.

6. $m\angle A = m\angle B$.
   Reason: Base angles of an isosceles triangle are congruent.

7. $m\angle B + m\angle B = m\angle AOC$.
   Reason: Substituting $m\angle A = m\angle B$ from Step 6.

8. $m\angle B = \frac{1}{2}m\angle AOC$.
   Reason: Divide both sides by 2.

9. $m\overarc{AC} = m\angle AOC$.
   Reason: Central angle theorem.

10. Substitution Property: $m\angle B = \frac{1}{2}m\overarc{AC}$.
    Reason: Substitution from Step 8.

Let me know if you want further clarification or step-by-step details! Here are 5 related questions to practice:

1. What is the relationship between a central angle and an inscribed angle intercepting the same arc?
2. How do you prove that opposite angles in an inscribed quadrilateral sum to 180°?
3. How can the diameter of a circle help identify inscribed right triangles?
4. Explain why $m\angle ABC = m\angle ADC$ when both intercept the same arc.
5. Why do all radii of a circle have equal lengths?

Tip: Always visualize circle properties like radii, diameters, and intercepted arcs to simplify proofs and problems!