1. Problem 9: Find the perimeter of triangle $\triangle MNP$ with sides $MN=22$, $MP=25$, and $NP=x+4$. Given angles $\angle M=55^\circ$, $\angle N=34^\circ$. 2. Use the triangle angle sum rule: $\angle M + \angle N + \angle P = 180^\circ$. 3. Calculate $\angle P$: $$\angle P = 180^\circ - 55^\circ - 34^\circ = 91^\circ$$ 4. Use the Law of Sines to find $x+4 = NP$: $$\frac{NP}{\sin \angle M} = \frac{MP}{\sin \angle N}$$ $$\Rightarrow \frac{x+4}{\sin 55^\circ} = \frac{25}{\sin 34^\circ}$$ 5. Solve for $x+4$: $$x+4 = \frac{25 \sin 55^\circ}{\sin 34^\circ}$$ Calculate values: $$\sin 55^\circ \approx 0.8192, \quad \sin 34^\circ \approx 0.5592$$ $$x+4 = \frac{25 \times 0.8192}{0.5592} \approx 36.62$$ 6. Find $x$: $$x = 36.62 - 4 = 32.62$$ 7. Calculate perimeter: $$P = MN + NP + MP = 22 + 36.62 + 25 = 83.62$$ --- 1. Problem 11: Find missing angles $m \angle 1, m \angle 2, m \angle 3, m \angle 4, m \angle 5$ in a right triangle with one angle $35^\circ$. 2. Sum of angles in triangle is $180^\circ$. Given one angle is $35^\circ$, and one right angle $90^\circ$. 3. Calculate third angle: $$m \angle 3 = 180^\circ - 90^\circ - 35^\circ = 55^\circ$$ 4. Assign angles: - $m \angle 1 = 90^\circ$ (right angle) - $m \angle 2 = 35^\circ$ - $m \angle 3 = 55^\circ$ 5. For $m \angle 4$ and $m \angle 5$, assuming they are exterior or other angles, if not specified, cannot determine from given data. --- 1. Problem 13: Find $x$ given two equal angles in an isosceles triangle: $$(5x - 24)^\circ = (8x + 9)^\circ$$ 2. Set equation: $$5x - 24 = 8x + 9$$ 3. Rearrange: $$5x - 8x = 9 + 24$$ $$-3x = 33$$ 4. Solve for $x$: $$x = \frac{33}{-3} = -11$$ --- 1. Problem 15: Find $m \angle ACB$ in quadrilateral with angles: $$(11x - 2)^\circ, (6x + 13)^\circ, 62^\circ$$ 2. Sum of interior angles in quadrilateral is $360^\circ$. 3. Sum given angles plus $m \angle ACB$: $$(11x - 2) + (6x + 13) + 62 + m \angle ACB = 360$$ 4. Simplify: $$17x + 73 + m \angle ACB = 360$$ 5. Solve for $m \angle ACB$: $$m \angle ACB = 360 - 17x - 73 = 287 - 17x$$ 6. To find numeric value, need $x$. If $x$ unknown, express answer as $m \angle ACB = 287 - 17x$. --- Final answers: - Problem 9 perimeter $\approx 83.62$ - Problem 11 angles $m \angle 1=90^\circ, m \angle 2=35^\circ, m \angle 3=55^\circ$ - Problem 13 $x = -11$ - Problem 15 $m \angle ACB = 287 - 17x$ (in terms of $x$)