1. **Problem 4: Which graph shows a line with an undefined slope?** A line with an undefined slope is a vertical line because the change in $x$ is zero, making the slope formula $m=\frac{\Delta y}{\Delta x}$ undefined. - Option b) has points (-1,2) and (-1,-3), which form a vertical line at $x=-1$. **Answer: b)** 2. **Problem 25: Find the slope of the line through points (-2,2) and (1,-2).** Formula for slope: $$m=\frac{y_2 - y_1}{x_2 - x_1}$$ Calculate: $$m=\frac{-2 - 2}{1 - (-2)}=\frac{-4}{1+2}=\frac{-4}{3}$$ **Slope is $-\frac{4}{3}$.** 3. **Problem 26: Write the equation in slope-intercept form for the line crossing the origin going upward diagonally.** Since it crosses the origin $(0,0)$, $b=0$. The slope $m$ is positive; assume $m=1$ for a 45° diagonal line. Equation: $$y=mx+b = y=1\cdot x + 0 = x$$ **Equation: $y=x$.** 4. **Problem 27: Write equation for line with slope $\frac{1}{2}$ passing through (1,2).** Use point-slope form: $$y - y_1 = m(x - x_1)$$ Substitute: $$y - 2 = \frac{1}{2}(x - 1)$$ Simplify: $$y = \frac{1}{2}x - \frac{1}{2} + 2 = \frac{1}{2}x + \frac{3}{2}$$ **Equation: $y=\frac{1}{2}x + \frac{3}{2}$.** 5. **Problem 28: What type of angles are $\angle 8$ and $\angle 13$ formed by parallel lines $m,n$ and transversal $t$?** Angles on opposite sides of the transversal and inside the parallel lines are alternate interior angles. **Answer: Alternate interior angles.** 6. **Problem 29: Name a pair of consecutive interior angles.** Consecutive interior angles lie on the same side of the transversal and inside the parallel lines. Options: - a) $\angle 10$ and $\angle 8$ (same side, inside) - correct - b) $\angle 10$ and $\angle 9$ (not both interior) - c) $\angle 6$ and $\angle 7$ (not consecutive interior) - d) $\angle 4$ and $\angle 6$ (not consecutive interior) **Answer: a)** 7. **Problem 30: If two lines are cut by a transversal and ______ angles are congruent, then the lines are parallel.** Correct answer: d) corresponding angles 8. **Problem 31: Find $x$ so $s \parallel t$ given angles $18x - 22$ and $22x + 2$ are consecutive interior angles.** Consecutive interior angles are supplementary: $$ (18x - 22) + (22x + 2) = 180 $$ Simplify: $$ 18x - 22 + 22x + 2 = 180 $$ $$ 40x - 20 = 180 $$ $$ 40x = 200 $$ $$ x = \frac{200}{40} = 5 $$ **Value of $x$ is 5.** 9. **Problem 32: Classify $\triangle FGH$ with vertices $F(-1,-5)$, $G(-3,4)$, $H(5,2)$ by its sides.** Calculate side lengths using distance formula: $$d=\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ - $FG = \sqrt{(-3 + 1)^2 + (4 + 5)^2} = \sqrt{(-2)^2 + 9^2} = \sqrt{4 + 81} = \sqrt{85}$ - $GH = \sqrt{(5 + 3)^2 + (2 - 4)^2} = \sqrt{8^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68}$ - $FH = \sqrt{(5 + 1)^2 + (2 + 5)^2} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85}$ Sides $FG$ and $FH$ are equal, $GH$ is different. **Triangle is isosceles.** 10. **Problem 33: Find $x$ in triangle with angles $27^\circ$, $18^\circ$, and $x$.** Sum of angles in triangle: $$27 + 18 + x = 180$$ $$45 + x = 180$$ $$x = 180 - 45 = 135$$ **$x = 135^\circ$.** 11. **Problem 34: Which pair of triangles shows $\triangle MNP \cong \triangle RST$ by AAS Theorem?** AAS requires two angles and a non-included side congruent. Option d) shows side and angle congruences consistent with AAS. **Answer: d)**