1. **Problem:** Name two lines or segments that lie in plane ADH. The plane ADH contains points A, D, and H. Lines or segments lying in this plane must connect these points. - Line DA connects points D and A. - Line AH connects points A and H. Among the options, only **DA and HG** is incorrect because HG is not in plane ADH. Correct answer: **a) DC and DA** is incorrect because DC is not in plane ADH. Correct answer: **d) EH and AE** is incorrect because EH is not in plane ADH. Correct answer: **c) AB and HD** is incorrect because AB is not in plane ADH. Correct answer: **b) DA and HG** is incorrect because HG is not in plane ADH. So the correct lines in plane ADH are **DA and AH** but since AH is not an option, the closest correct pair is **DA and DC** if DC lies in the base plane which is different from ADH. Since none of the options exactly match DA and AH, the best answer is **a) DC and DA** assuming DC lies in the base plane ADH. 2. **Problem:** Which of the following best describes ∠ AGB? Given points A, G, B on a line, angle AGB is formed at G between points A and B. Since A, G, B are collinear, angle AGB is a straight angle. Answer: **d) straight** 3. **Problem:** If $m \angle 1 = 45^\circ$ and $\angle FGE$ is a right angle, find $m \angle AGE$. - $\angle FGE = 90^\circ$ - $\angle 1 = 45^\circ$ Since $\angle AGE$ is adjacent to $\angle 1$ and $\angle FGE$, and these three angles form a straight line, their sum is $180^\circ$. Calculate: $$m \angle AGE = 180^\circ - 90^\circ - 45^\circ = 45^\circ$$ 4. **Problem:** Find measures of supplementary angles $\angle M$ and $\angle N$ given: $$m \angle M = 6x + 3$$ $$m \angle N = 2x - 7$$ Supplementary angles sum to $180^\circ$: $$6x + 3 + 2x - 7 = 180$$ $$8x - 4 = 180$$ $$8x = 184$$ $$x = \frac{184}{8} = 23$$ Find measures: $$m \angle M = 6(23) + 3 = 138 + 3 = 141^\circ$$ $$m \angle N = 2(23) - 7 = 46 - 7 = 39^\circ$$ 5. **Problem:** Identify the polygon shown (five-sided irregular polygon). - A polygon with 5 sides is a pentagon. - Since it is irregular and has a concave shape, the correct description is **a) concave**. 6. **Problem:** The intersection of two planes could be a: - Two planes intersect in a line. Answer: **b) line** 7. **Problem:** Find distance $PQ$ for points $P(2,7)$ and $Q(-4,2)$. Use distance formula: $$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-4 - 2)^2 + (2 - 7)^2} = \sqrt{(-6)^2 + (-5)^2} = \sqrt{36 + 25} = \sqrt{61} \approx 7.8$$ 8. **Problem:** Find midpoint $R$ of segment $QS$ with endpoints $Q(3,-5)$ and $S(-3,6)$. Midpoint formula: $$R = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{3 + (-3)}{2}, \frac{-5 + 6}{2} \right) = (0, 0.5)$$ 9. **Problem:** The supplement of an angle is 30 less than four times the complement of the angle. Find the angle. Let the angle be $x$. - Supplement: $180 - x$ - Complement: $90 - x$ Equation: $$180 - x = 4(90 - x) - 30$$ $$180 - x = 360 - 4x - 30$$ $$180 - x = 330 - 4x$$ $$-x + 4x = 330 - 180$$ $$3x = 150$$ $$x = 50$$ Answer: **50** 10. **Problem:** If $DF = 15$ and $DE = 6$, find $EF$ if $E$ is between $D$ and $F$. Since $E$ is between $D$ and $F$: $$DF = DE + EF$$ $$15 = 6 + EF$$ $$EF = 15 - 6 = 9$$ 11. **Problem:** Find perimeter of a regular pentagon with side length 57 cm. Perimeter formula: $$P = 5 \times 57 = 285$$ 12. **Problem:** Given $x^2 = 4$, conjecture: $x = 4$ or $x = -4$. Determine if true or false. Solve: $$x^2 = 4$$ $$x = \pm 2$$ Conjecture is false because $x = 4$ or $-4$ is incorrect. **Final answers:** 1) a) DC and DA 2) d) straight 3) $45^\circ$ 4) $m \angle M = 141^\circ$, $m \angle N = 39^\circ$ 5) a) concave 6) b) line 7) $7.8$ 8) $(0, 0.5)$ 9) $50$ 10) $9$ 11) $285$ 12) False