Question

module 2 topic 1 assessment review

1. consider the circle on the right:

a. identify a minor arc.
b. identify a major arc.
c. identify a semicircle.
d. identify 2 inscribed angles.
e. identify 2 central angles.
f. if \\( \angle adf \\) is \\( 57^\circ \\), what is the measure of \\( \widehat{af} \\).

Explanation:

Step1: Recall definitions

• Minor arc: An arc that is less than a semicircle.
• Major arc: An arc that is greater than a semicircle.
• Semicircle: An arc that is exactly half of a circle.
• Inscribed angle: An angle whose vertex is on the circle and whose sides contain chords of the circle.
• Central angle: An angle whose vertex is at the center of the circle.
• For part f, use the central - angle - inscribed - angle relationship: The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.

Step2: Answer each part

• a. A minor arc could be $\overset{\frown}{AB}$ (answers may vary).
• b. A major arc could be $\overset{\frown}{AED}$ (answers may vary).
• c. A semicircle could be $\overset{\frown}{ABD}$ (answers may vary).
• d. Two inscribed angles could be $\angle AFD$ and $\angle AED$ (answers may vary).
• e. Two central angles could be $\angle ACD$ and $\angle FCD$ (answers may vary).
• f. Given that $\angle ADF$ is an inscribed angle and it subtends arc $\overset{\frown}{AF}$. Let the central angle subtending arc $\overset{\frown}{AF}$ be $\angle ADF$. Using the formula $m\angle ADF=\frac{1}{2}m\overset{\frown}{AF}$. If $m\angle ADF = 57^{\circ}$, then $m\overset{\frown}{AF}=2\times57^{\circ}=114^{\circ}$.

Answer:

a. $\overset{\frown}{AB}$ (example)
b. $\overset{\frown}{AED}$ (example)
c. $\overset{\frown}{ABD}$ (example)
d. $\angle AFD$ and $\angle AED$ (example)
e. $\angle ACD$ and $\angle FCD$ (example)
f. $114^{\circ}$