Question

(5 points each)a) 119.5°b) 121c) 270°d) 238°14) in circle o, $overline{rs}$ is a diameter. if the measure of arc pr =64°, what is m∠spr?a) 64°b) 58°c) 116°d) 32°) what is the measure of the interior angle?a) 152°b) 36.5°c) 79°d) 115.5°f marc acb = 230°, find m∠dab?a) 130°b) 230°c) 115°d) 65°d is an inscribed quadrilateral. find the measure of ∠b and justification for your answer.a) m∠b = 92°; consecutive angles are supplementary.b) m∠b = 75°; opposite angles are supplementary.c) m∠b = 105°; opposite angles are congruent.d) m∠b = 88°; consecutive angles are congruent.

Explanation:

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Question 14

Step1: Identify diameter arc measure

Arc $PSR$ is a semicircle, so $m\overset{\frown}{PSR}=180^\circ$.

Step2: Calculate arc $SP$

$m\overset{\frown}{SP}=180^\circ - m\overset{\frown}{PR}=180^\circ - 64^\circ=116^\circ$

Step3: Find inscribed angle $\angle SPR$

Inscribed angle = $\frac{1}{2}$ its intercepted arc:
$m\angle SPR=\frac{1}{2}m\overset{\frown}{PR}=\frac{1}{2}\times64^\circ=32^\circ$

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Question (no number, interior angle)

Step1: Identify intercepted arcs

The interior angle intercepts arcs of $79^\circ$ and $152^\circ$.

Step2: Calculate interior angle measure

Interior angle = $\frac{1}{2}$ sum of intercepted arcs:
$m\angle=\frac{1}{2}(79^\circ + 152^\circ)=\frac{1}{2}\times231^\circ=115.5^\circ$

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Question (find $m\angle DAB$)

Step1: Calculate arc $AB$

Total circle = $360^\circ$, so $m\overset{\frown}{AB}=360^\circ - 230^\circ=130^\circ$

Step2: Find tangent-chord angle $\angle DAB$

Tangent-chord angle = $\frac{1}{2}$ its intercepted arc:
$m\angle DAB=\frac{1}{2}m\overset{\frown}{AB}=\frac{1}{2}\times130^\circ=65^\circ$

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Question (inscribed quadrilateral $\angle B$)

Step1: Recall inscribed quadrilateral rule

Opposite angles of an inscribed quadrilateral are supplementary (sum to $180^\circ$).

Step2: Calculate $m\angle B$

From handwritten work, opposite angle to $\angle B$ is $75^\circ$, so:
$m\angle B=180^\circ - 75^\circ=105^\circ$

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Answer:

1. D) $32^\circ$

Interior angle question: D) $115.5^\circ$
$m\angle DAB$ question: D) $65^\circ$
Inscribed quadrilateral question: C) $m\angle B = 105^\circ$; Opposite angles are supplementary.