Question

use the image to the right to answer questions #2-4.

1. solve for x and y.
2. what is the $m\widehat{ac}$?
3. if the $m\widehat{ad}=112^o$, then find the $m\widehat{cd}$.

use the image to the right for questions #5-6.

1. what is the measure of $\angle adb$?
2. if $\widehat{dc}=(7x + 6)^o$, then solve for x.

use the figure to answer #7-8.

1. find the measure of $\widehat{bc}$.

find the measure of $\angle cab$.

Explanation:

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Questions 2-4 (Cyclic Quadrilateral ABCD)

Step1: Solve for y (cyclic quad sum)

Sum of quad angles = $360^\circ$
$y = 360 - (94 + 110 + x)$; first find x: inscribed angle subtended by $\overset{\frown}{BD}$: $x = 180 - 110 = 70^\circ$
$y = 360 - (94 + 110 + 70) = 86^\circ$

Step2: Find $m\overset{\frown}{AC}$

Inscribed angle $\angle ABC = 94^\circ$, so arc $AC = 2 \times 94 = 188^\circ$

Step3: Find $m\overset{\frown}{CD}$

Total circle = $360^\circ$. $m\overset{\frown}{AD}=112^\circ$, $m\overset{\frown}{AC}=188^\circ$, so $m\overset{\frown}{CD} = 360 - 112 - (360 - 188) = 76^\circ$
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Questions 5-6 (Circle with $\overset{\frown}{AB}=76^\circ$, $\angle DBC=45^\circ$)

Step4: Find $\angle ADB$

$\angle ADB$ subtends $\overset{\frown}{AB}$, so $\angle ADB = \frac{1}{2} \times 76 = 38^\circ$

Step5: Solve for x ($\overset{\frown}{DC}=7x+6$)

$\angle DBC$ subtends $\overset{\frown}{DC}$, so $\overset{\frown}{DC}=2 \times 45 = 90^\circ$
$7x + 6 = 90$
$7x = 84$
$x = 12$
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Questions 7-8 (Semicircle, $\overset{\frown}{AC}=108^\circ$)

Step6: Find $m\overset{\frown}{BC}$

Semicircle arc = $180^\circ$, so $\overset{\frown}{BC}=180 - 108 = 72^\circ$

Step7: Find $\angle CAB$

$\angle CAB$ subtends $\overset{\frown}{BC}$, so $\angle CAB = \frac{1}{2} \times 72 = 36^\circ$

Answer:

1. Question 2: $x=70^\circ$, $y=86^\circ$
2. Question 3: $m\overset{\frown}{AC}=188^\circ$
3. Question 4: $m\overset{\frown}{CD}=76^\circ$
4. Question 5: $m\angle ADB=38^\circ$
5. Question 6: $x=12$
6. Question 7: $m\overset{\frown}{BC}=72^\circ$
7. Question 8: $m\angle CAB=36^\circ$