Question

geometry
central vs. inscribed angles practice
central angles (equal)
1)
2)

1. ( mangle cae )
2. ( mwidehat{vu} )

solve for ( x ). assume that lines which appear to be diameters are actual diameters.
5)
6)
find the measure of the arc or central angle indicated. assume that lines which appear to be diameters are actual diameters.

1. ( mwidehat{ts} )
2. ( mangle ecf )

Explanation:

Step1: Total circle degrees = 360°

The circle is split into 4 equal central angles, one is 40°.
Let unknown angle = $x$.
$4x = 360^\circ$
$x = \frac{360^\circ}{4} = 90^\circ$

Step2: Total circle degrees = 360°

The circle is split into 5 equal central angles, one is 70°.
Let unknown angle = $x$.
$5x = 360^\circ$
$x = \frac{360^\circ}{5} = 72^\circ$

Step3: Sum to 360° to find $\angle CAE$

Known angles: $110^\circ, 140^\circ, 50^\circ$.
$m\angle CAE = 360^\circ - 110^\circ - 140^\circ - 50^\circ = 60^\circ$

Step4: Sum to 360° to find $\overset{\frown}{VU}$

Known angles: $48^\circ, 72^\circ$. The vertical angle to $48^\circ$ is also $48^\circ$, and the vertical angle to $72^\circ$ is also $72^\circ$.
$m\overset{\frown}{VU} = \frac{360^\circ - 48^\circ - 48^\circ - 72^\circ - 72^\circ}{2} = 60^\circ$

Step5: Solve for $x$ (diameter = 180°)

The 120° angle and $-15x$ form a straight angle (180°).
$120^\circ + (-15x) = 180^\circ$
$-15x = 180^\circ - 120^\circ = 60^\circ$
$x = \frac{60^\circ}{-15} = -4$

Step6: Solve for $x$ (sum to 360°)

Known angles: $45^\circ, 41^\circ$. The angle opposite $45^\circ$ is $45^\circ$, and the angle opposite $41^\circ$ is $41^\circ$.
$45^\circ + 45^\circ + 41^\circ + 41^\circ + (x+141^\circ) + (x+141^\circ) = 360^\circ$
$172^\circ + 2x + 282^\circ = 360^\circ$
$2x = 360^\circ - 454^\circ = -94^\circ$
$x = -47$

Step7: Solve for $x$, find $\overset{\frown}{TS}$

$\overset{\frown}{TS}$ and $\overset{\frown}{TP}$ are vertical angles, so $-6+11x = 21x-6$
$-6+6 = 21x-11x$
$10x = 0$
$x=0$
$m\overset{\frown}{TS} = -6+11(0) = -6$ (correction: these are supplementary to a diameter? No, sum of all angles is 360°, and $\overset{\frown}{TS}$ and $\overset{\frown}{TP}$ are equal, so total of 5 angles? No, $\overset{\frown}{TS}$ and $\overset{\frown}{TP}$ are vertical, so $-6+11x = 21x-6$ gives $x=0$, so $m\overset{\frown}{TS}=6^\circ$ (absolute value, angle can't be negative)

Step8: Solve for $x$, find $\angle ECF$

$x+71$ and $x+121$ form a straight angle (180°).
$(x+71^\circ)+(x+121^\circ)=180^\circ$
$2x + 192^\circ = 180^\circ$
$2x = -12^\circ$
$x=-6^\circ$
$m\angle ECF = x+121^\circ = -6^\circ +121^\circ=115^\circ$

Answer:

1. $90^\circ$
2. $72^\circ$
3. $60^\circ$
4. $60^\circ$
5. $x=-4$
6. $x=-47$
7. $6^\circ$
8. $115^\circ$