11. find the measure of $\angle pqr$. list in sequence with reasons the angles you had to find along the way to determine the required angle.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Triangle Angle Sum Theorem", "Exterior Angle Theorem" ], "new_concepts": [ "Angle Bisector Properties", "Perpendicular Lines" ], "current_concepts": [ "Triangle Angle Sum Theorem", "Exterior Angle Theorem", "Angle Bisector Properties", "Perpendicular Lines" ] } ## Find the measure of angle MVT Using the *Triangle Angle Sum Theorem* and *Exterior Angle Theorem* knowledge points $$ \begin{aligned} &\angle VPM = 180^\circ - 80^\circ = 100^\circ\ &\angle MVP = 180^\circ - 50^\circ - 100^\circ = 30^\circ \end{aligned} $$ ## Find the bisected angle measures Using the *Angle Bisector Theorem* knowledge point $$ \begin{aligned} &\angle MVS = \angle SVR = \frac{1}{2} \angle MVP = 15^\circ\ &\angle TVS = \angle MVS = 15^\circ \end{aligned} $$ ## Find the measure of angle TSV Using the *Triangle Angle Sum Theorem* knowledge point $$ \begin{aligned} &\angle VSN = 90^\circ\ &\angle TSV = 180^\circ - 90^\circ = 90^\circ \end{aligned} $$ ## Find the measure of angle MTS Using the *Triangle Angle Sum Theorem* knowledge point $$ \begin{aligned} &\angle MTS = 180^\circ - \angle TSV - \angle TVS\ &\angle MTS = 180^\circ - 90^\circ - 15^\circ = 75^\circ \end{aligned} $$ ## Find the measure of angle PQR Using the *Triangle Angle Sum Theorem* knowledge point $$ \begin{aligned} &\angle MTQ = 75^\circ\ &\angle PMT = 50^\circ\ &\angle PQR = 180^\circ - \angle MTQ - \angle PMT\ &\angle PQR = 180^\circ - 75^\circ - 50^\circ = 55^\circ \end{aligned} $$ To find the measure of $\angle PQR$, the sequence of angles determined along with their reasons is: 1. **$\angle VPM = 100^\circ$**: Straight Angle Theorem at point $P$ ($180^\circ - 80^\circ$). 2. **$\angle MVP = 30^\circ$**: Triangle Angle Sum Theorem in $\triangle MVP$ ($180^\circ - 50^\circ - 100^\circ$). 3. **$\angle MVS = \angle SVR = 15^\circ$**: Given that $VS$ bisects $\angle MVP$. 4. **$\angle TSV = 90^\circ$**: Straight Angle Theorem at point $S$ since $VN \perp TQ$. 5. **$\angle MTS = 75^\circ$**: Triangle Angle Sum Theorem in $\triangle TSV$ ($180^\circ - 90^\circ - 15^\circ$). 6. **$\angle PQR = 55^\circ$**: Triangle Angle Sum Theorem in $\triangle MTQ$ ($180^\circ - 50^\circ - 75^\circ$). The final measure is: $$\angle PQR = 55^\circ$$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Angle Sum Theorem" ] }

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Explanation:

Find the measure of angle MVT

Find the bisected angle measures

Find the measure of angle TSV

Find the measure of angle MTS

Find the measure of angle PQR

Answer: