Step 1: Recall the Inscribed Angle Theorem: An inscribed angle is half the measure of the central angle that subtends the same arc

Step 2: Apply the theorem: Inscribed angle = (1/2) × Central angle Inscribed angle = (1/2) × 80° Inscribed angle = 40°

Step 3: Understand why: The inscribed angle has its vertex on the circle The central angle has its vertex at the center They both intercept the same arc

Answer: The inscribed angle measures 40°

Use the Inscribed Angle Theorem:

$\text{Inscribed angle} = \frac{1}{2} \times \text{arc}$

$= \frac{1}{2} \times 80°$

$= 40°$

Answer: The inscribed angle is $40°$

Step 1: Recall the relationship: Inscribed angle = (1/2) × intercepted arc

Step 2: Set up equation: 55° = (1/2) × arc

Step 3: Solve for the arc: arc = 2 × 55° arc = 110°

Step 4: Verify: Inscribed angle = 110°/2 = 55° ✓

Answer: The intercepted arc measures 110°

A semicircle is an arc of 180°.

By the Inscribed Angle Theorem: $\text{Inscribed angle} = \frac{1}{2} \times 180° = 90°$

Answer: The angle is $90°$ (a right angle)

Step 1: Identify the intercepted arc: Inscribed angle ABC has vertex at B It intercepts arc AC (the short way)

Step 2: Find the short arc AC: Total circle = 360° Arc ABC (long way) = 250° Arc AC (short way) = 360° - 250° = 110°

Step 3: Apply Inscribed Angle Theorem: Inscribed angle = (1/2) × intercepted arc Angle ABC = (1/2) × 110° Angle ABC = 55°

Step 4: Important note: The inscribed angle intercepts the arc that does NOT contain the vertex

Answer: Angle ABC = 55°

Step 1: Recall inscribed quadrilateral property: Opposite angles in an inscribed quadrilateral (cyclic quadrilateral) are supplementary

Step 2: Find angle C (opposite to angle A): Angle A + Angle C = 180° 110° + Angle C = 180° Angle C = 70°

Step 3: Find angle D (opposite to angle B): Angle B + Angle D = 180° 75° + Angle D = 180° Angle D = 105°

Step 4: Verify all angles sum to 360°: 110° + 75° + 70° + 105° = 360° ✓

Step 5: Verify opposite pairs: A + C = 110° + 70° = 180° ✓ B + D = 75° + 105° = 180° ✓

Answer: Angle C = 70°, Angle D = 105°

In an inscribed quadrilateral, opposite angles are supplementary.

Angles A and C are opposite: $\angle A + \angle C = 180°$

$75 + (2x + 15) = 180$

$2x + 90 = 180$

$2x = 90$

$x = 45$

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

Answer: $x = 45$

Step 1: Identify what we know: AB is a diameter C is a point on the circle Need to prove: angle ACB = 90°

Step 2: Understand the intercepted arc: Since AB is a diameter, it divides the circle into two semicircles Arc ACB (going through C) is a semicircle = 180°

Step 3: Apply Inscribed Angle Theorem: Inscribed angle ACB intercepts arc AB Since AB is a diameter, arc AB = 180° (semicircle)

Angle ACB = (1/2) × 180° Angle ACB = 90°

Step 4: State the theorem: This proves Thales' Theorem: "An angle inscribed in a semicircle is a right angle"

Step 5: Why this works: Any triangle inscribed in a semicircle with the diameter as one side must be a right triangle

Step 6: Conclusion: Angle ACB = 90° ✓

This theorem is extremely useful: If you know AB is a diameter and C is any other point on the circle, triangle ABC is always a right triangle with the right angle at C

Answer: Angle ACB = 90° (proved by Inscribed Angle Theorem and the fact that a diameter creates a 180° arc)