Question

central angles homework

1. identify and name each of the following from ⊙g. be sure to use the correct notation. bd is a diameter.

a. two different central angles
b. a minor arc
c. a major arc
d. a semicircle
e. two different chords
f. the central angle subtended by ( overline{ad} )
find each measure.
it is a diameter.

1. ( mangle jhk ), ( mwidehat{jk} )

ru & qt are diameters.

1. ( mwidehat{qs} ), ( mangle rot )

Explanation:

Step1: Solve Question 1 (Circle O)

Part a: Name 2 central angles

Central angles have vertex at center $O$.
Examples: $\angle AOB$, $\angle BOE$

Part b: Name a minor arc

Minor arc < 180°, 2 points: $\overset{\frown}{AB}$

Part c: Name a major arc

Major arc > 180°, 3 points: $\overset{\frown}{ABE}$

Part d: Name a semicircle

Semicircle = 180°, uses diameter $BD$: $\overset{\frown}{BCD}$

Part e: Name 2 different chords

Chords connect 2 circle points: $\overline{AB}$, $\overline{AE}$

Part f: Central angle for $\overset{\frown}{AD}$

Vertex at $O$, so $\angle AOD$

Step2: Solve Question 2 (Circle H)

Find $m\overset{\frown}{JK}$

$\angle JHL=90^\circ$, $\angle KHL=30^\circ$. Right angle $\angle JHI=90^\circ$, so $\angle JHK=180^\circ-90^\circ-30^\circ=60^\circ$. Arc = central angle: $m\overset{\frown}{JK}=60^\circ$

Find $m\angle K$

$\overline{HJ}=\overline{HK}$ (radii), so $\triangle HJK$ is isosceles. $\angle HJK=\angle HKJ$. Sum angles = 180°:
$\angle HKJ=\frac{180^\circ-60^\circ}{2}=60^\circ$

Step3: Solve Question 3 (Circle P)

Find $m\overset{\frown}{QS}$

$\overline{RU}$ is diameter, so $\angle RPU=180^\circ$. $\angle UPT=47^\circ$, $\angle TPS=55^\circ$, so $\angle RPS=180^\circ-47^\circ-55^\circ=78^\circ$. Vertical angle $\angle QPU=\angle RPS=78^\circ$, so $m\overset{\frown}{QS}=78^\circ+55^\circ=133^\circ$

Find $m\angle TQT$ (corrected to $\angle QPT$)

$\overline{QT}$ is diameter, so $\angle QPT=180^\circ$

Answer:

1.
a. $\angle AOB$, $\angle BOE$
b. $\overset{\frown}{AB}$
c. $\overset{\frown}{ABE}$
d. $\overset{\frown}{BCD}$
e. $\overline{AB}$, $\overline{AE}$
f. $\angle AOD$

2.
$m\overset{\frown}{JK}=60^\circ$
$m\angle K=60^\circ$

3.
$m\overset{\frown}{QS}=133^\circ$
$m\angle QPT=180^\circ$