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" ], "current_concepts": [ "Triangle Angle Sum Theorem", "Exterior Angle Theorem", "Angle Bisector Properties" ] } ## Identify the given angle measures and geometric markings We are given the following information from the diagram: - Points $M$, $N$, $P$, and $Q$ lie on a straight line in that order. - In $\triangle MVP$, the angle at vertex $M$ is $m\angle VMP = 50^\circ$. - The exterior angle at vertex $P$ is $m\angle VPQ = 80^\circ$. - The line segment $VN$ is an angle bisector of $\angle MVP$, indicated by the matching tick marks on $\angle MVN$ and $\angle NV P$. Thus, $m\angle MVN = m\angle NVP$. - The line segment $TQ$ intersects $VN$ at point $S$ and $VP$ at point $R$. - The intersection of $VN$ and $TQ$ is perpendicular, meaning $m\angle VSQ = m\angle VST = 90^\circ$ (and similarly for the other angles at intersection $S$). ## Calculate the angles of $\triangle MVP$ Using the *Exterior Angle Theorem* knowledge point $$ \begin{aligned} m\angle VPQ &= m\angle VMP + m\angle MVP \ 80^\circ &= 50^\circ + m\angle MVP \ m\angle MVP &= 30^\circ \end{aligned} $$ ## Determine the bisected angle measures Since $VN$ bisects $\angle MVP$: $$ \begin{aligned} m\angle MVN = m\angle NVP = \frac{m\angle MVP}{2} = \frac{30^\circ}{2} = 15^\circ \end{aligned} $$ ## Find the angles in right triangle $\triangle VSG$ to determine $\angle PQR$ Using the *Triangle Angle Sum Theorem* knowledge point In right $\triangle VSG$ (where $G$ is on the line $VN$, and $S$ is the intersection on $VN$): Let's look at $\triangle VSR$ where $R$ is on $VP$. Since $VN \perp TQ$, $\triangle VSR$ is a right triangle with $m\angle VSR = 90^\circ$. $$ \begin{aligned} m\angle VRS &= 90^\circ - m\angle SVR \ &= 90^\circ - 15^\circ = 75^\circ \end{aligned} $$ Since $\angle PRQ$ and $\angle VRS$ are vertical angles: $$ m\angle PRQ = m\angle VRS = 75^\circ $$ ## Calculate the final target angle $\angle PQR$ Using the *Triangle Angle Sum Theorem* knowledge point In $\triangle PRQ$, the sum of angles is $180^\circ$: $$ \begin{aligned} m\angle PQR + m\angle RPQ + m\angle PRQ &= 180^\circ \ m\angle PQR + 80^\circ + 75^\circ &= 180^\circ \ m\angle PQR + 155^\circ &= 180^\circ \ m\angle PQR &= 25^\circ \end{aligned} $$ To find the measure of $\angle PQR$, the sequence of angles determined along with their reasons is: 1. **$m\angle MVP = 30^\circ$** *Reason:* By the Exterior Angle Theorem applied to $\triangle MVP$, where the exterior angle $m\angle VPQ = m\angle VMP + m\angle MVP$. 2. **$m\angle NVP = 15^\circ$** *Reason:* Given that $VN$ is the angle bisector of $\angle MVP$, $m\angle NVP = \frac{1}{2} m\angle MVP$. 3. **$m\angle VRS = 75^\circ$** *Reason:* In right $\triangle VSR$ (since $VN \perp TQ$), the acute angles sum to $90^\circ$, so $m\angle VRS = 90^\circ - m\angle SVR$. 4. **$m\angle PRQ = 75^\circ$** *Reason:* Vertical angles are congruent ($\angle PRQ \cong \angle VRS$). 5. **$m\angle PQR = 25^\circ$** *Reason:* By the Triangle Angle Sum Theorem applied to $\triangle PRQ$, the interior angles sum to $180^\circ$. { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Angle Sum Theorem" ] }

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Explanation:

Identify the given angle measures and geometric markings

Calculate the angles of \(\triangle MVP\)

Determine the bisected angle measures

Find the angles in right triangle \(\triangle VSG\) to determine \(\angle PQR\)

Calculate the final target angle \(\angle PQR\)

Answer: