in the following diagram, determine the value of (x):

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Angle Bisector Theorem", "Triangle Congruence" ], "new_concepts": [], "current_concepts": [ "Angle Bisector Theorem", "Triangle Congruence", "Right Triangles" ] } ## Identify congruent triangles Using the **Triangle Congruence** knowledge point $$ \begin{aligned} &\text{Let the shared vertex on the right be } A.\ &\text{Let the vertex on the left be } P.\ &\text{Let the perpendicular projections on the upper and lower rays be } B \text{ and } C \text{ respectively.}\ &\text{In right triangles } \triangle ABP \text{ and } \triangle ACP:\ &AP = AP \quad (\text{Reflexive Property})\ &PB = PC \quad (\text{Given by tick marks})\ &\angle ABP = \angle ACP = 90^\circ \quad (\text{Given by right angle symbols})\ &\triangle ABP \cong \triangle ACP \quad (\text{Hypotenuse-Leg Congruence Theorem}) \end{aligned} $$ ## Apply angle bisector properties Using the **Angle Bisector Theorem** knowledge point $$ \begin{aligned} &\angle BAP = \angle CAP\ &15x - 2 = 33 \end{aligned} $$ ## Solve for x Using the **Angle Bisector Theorem** knowledge point $$ \begin{aligned} &15x = 35\ &x = \frac{35}{15} = \frac{7}{3} \end{aligned} $$ $x = \frac{7}{3}$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Angle Bisector Theorem" ] }

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Question

Explanation:

Identify congruent triangles

Apply angle bisector properties

Solve for x

Answer: