Corollaries

Inscribed angles that intercept the same arc are congruent.

An angle inscribed in a semicircle is a right angle (90°).

• Because it intercepts a 180° arc
• $\frac{180°}{2} = 90°$

Inscribed Quadrilateral

If a quadrilateral is inscribed in a circle:

Opposite angles are supplementary (sum to 180°)

Arc-Angle Relationships

• Central angle = arc measure
• Inscribed angle = $\frac{1}{2}$ arc measure
• Tangent-chord angle = $\frac{1}{2}$ arc measure

$= \frac{1}{2} \times 80°$

$= 40°$

Answer: The inscribed angle is $40°$

$2x + 90 = 180$

$2x = 90$

$x = 45$

Verify: $\angle C = 2(45) + 15 = 105°$, and $75° + 105° = 180°$ ✓

Answer: $x = 45$