in the diagram below, $\overline{xy}$ and $\overline{yz}$ are tangent to $\odot o$. what is the measure of $\widehat{xz}$?

a. $90^\circ$
b. $130^\circ$
c. $100^\circ$
d. $180^\circ$

Question

in the diagram below, $\overline{xy}$ and $\overline{yz}$ are tangent to $\odot o$. what is the measure of $\widehat{xz}$?

a. $90^\circ$
b. $130^\circ$
c. $100^\circ$
d. $180^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Angles Outside Circle Theorem", "Minor Arc" ], "new_concepts": [], "current_concepts": [ "Angles Outside Circle Theorem", "Minor Arc", "Major Arc", "Tangent Line" ] } ## Identify the given values and target unknown Using the *Angles Outside Circle Theorem* and *Minor Arc* knowledge points $$ \begin{aligned} &\text{Given: } m\angle Y = 50^\circ\ &\text{Given: } \text{Major arc } \widehat{XZ} = 230^\circ\ &\text{Target: } m\widehat{XZ} \text{ (minor arc)} \end{aligned} $$ ## Apply the Angles Outside Circle Theorem Using the *Angles Outside Circle Theorem* knowledge point $$ \begin{aligned} &m\angle Y = \frac{1}{2}(\text{Major arc } \widehat{XZ} - \text{Minor arc } \widehat{XZ})\ &50^\circ = \frac{1}{2}(230^\circ - m\widehat{XZ}) \end{aligned} $$ ## Solve for the measure of the minor arc Using the *Minor Arc* knowledge point $$ \begin{aligned} &100^\circ = 230^\circ - m\widehat{XZ}\ &m\widehat{XZ} = 230^\circ - 100^\circ\ &m\widehat{XZ} = 130^\circ \end{aligned} $$

A. $90^\circ$

B. $130^\circ$

C. $100^\circ$

D. $180^\circ$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Angles Outside Circle Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given values and target unknown

Apply the Angles Outside Circle Theorem

Solve for the measure of the minor arc

Answer: