Question: Q1 p = doc.add_paragraph() p.add_run("1. (a) ").bold = True p.add_run("In the diagram, MNPS is a quadrilateral. A line is drawn through N to cut SP at Q. Angle MNQ = 132\u00b0, angle SMN is twice angle MSQ and angle NPQ is twice angle QNP. If NP bisects the acute angle at N, find:") doc.add_picture(diagram1_path, width=Inches(5.4)) doc.paragraphs[-1].alignment = WD_ALIGN_PARAGRAPH.CENTER for text in ["i. Angle SQN.", "ii. Angle MSQ."]: p = doc.add_paragraph(text) p.paragraph_format.left_indent = Inches(0.35)

1. **Problem Statement:** In quadrilateral MNPS, a line through point N cuts SP at Q. Given: - $\angle MNQ = 132^\circ$ - $\angle SMN = 2 \times \angle MSQ$ - $\angle NPQ = 2 \times \angle QNP$ - $NP$ bisects the acute angle at $N$ Find: (i) $\angle SQN$ (ii) $\angle MSQ$ 2. **Understanding the problem and notation:** - Since $NP$ bisects the acute angle at $N$, let the acute angle at $N$ be $2x$, so each bisected angle is $x$. - Let $\angle QNP = y$, then $\angle NPQ = 2y$. - Let $\angle MSQ = z$, then $\angle SMN = 2z$. 3. **Using the given angles and relationships:** - From the quadrilateral and line properties, we analyze angles around point $N$ and on line $SP$. - Since $\angle MNQ = 132^\circ$, and $NP$ bisects the acute angle at $N$, the angles around $N$ must satisfy: $$\angle MNQ + \angle QNP + \angle NPQ = 180^\circ$$ Substitute $\angle MNQ = 132^\circ$, $\angle QNP = y$, $\angle NPQ = 2y$: $$132 + y + 2y = 180$$ $$132 + 3y = 180$$ $$3y = 48$$ $$y = 16^\circ$$ 4. **Find $\angle SQN$:** - $\angle SQN$ is supplementary to $\angle NPQ$ on line $SP$: $$\angle SQN + \angle NPQ = 180^\circ$$ Substitute $\angle NPQ = 2y = 32^\circ$: $$\angle SQN = 180 - 32 = 148^\circ$$ 5. **Find $\angle MSQ$:** - Given $\angle SMN = 2 \times \angle MSQ$, let $\angle MSQ = z$, so $\angle SMN = 2z$. - Since $\angle MNQ = 132^\circ$ and $\angle SMN = 2z$, angles around point $M$ in triangle $MSN$ satisfy: $$\angle SMN + \angle MSQ + \angle MNQ = 180^\circ$$ Substitute: $$2z + z + 132 = 180$$ $$3z = 48$$ $$z = 16^\circ$$ **Final answers:** - (i) $\angle SQN = 148^\circ$ - (ii) $\angle MSQ = 16^\circ$