1. Problem 8: Find $m\angle K$. 2. Problem 9: Find $m\angle Q$. 3. Problem 10: Given $KP=4$ and $PM=7$, find the length $KM$. 4. Problem 11: Given $XZ=18$ and $PY=3$, find the length $PW$. 5. Problem 12: Verify if quadrilateral $ABCD$ with vertices $A(-2,5)$, $B(-3,1)$, $C(6,1)$, $D(3,5)$ is a trapezoid and determine if it is isosceles. 6. Problem 13: Verify if quadrilateral $JKLM$ with vertices $J(-4,-6)$, $K(6,2)$, $L(1,3)$, $M(-4,-1)$ is a trapezoid and determine if it is isosceles. 7. Problem 14: Verify if quadrilateral $QRST$ with vertices $Q(2,5)$, $R(-2,1)$, $S(-1,-6)$, $T(9,4)$ is a trapezoid and determine if it is isosceles. 8. Problem 15: Verify if quadrilateral $WXYZ$ with vertices $W(-5,-1)$, $X(-2,2)$, $Y(3,1)$, $Z(5,-3)$ is a trapezoid and determine if it is isosceles. --- **Step 1: Solve Problem 8: Find $m\angle K$.** Since no additional information is given about angle $K$, and the figure is not fully described, we cannot calculate $m\angle K$ without more data. --- **Step 2: Solve Problem 9: Find $m\angle Q$.** Similarly, no data is provided to calculate $m\angle Q$. --- **Step 3: Solve Problem 10: Given $KP=4$ and $PM=7$, find $KM$.** 1. The segment $KM$ is composed of $KP$ and $PM$. 2. By the Segment Addition Postulate: $$KM = KP + PM$$ 3. Substitute values: $$KM = 4 + 7 = 11$$ **Answer:** $m(KM) = 11$ --- **Step 4: Solve Problem 11: Given $XZ=18$ and $PY=3$, find $PW$.** Without additional information about the relationship between $PW$, $PY$, and $XZ$, we cannot determine $PW$. --- **Step 5: Solve Problem 12: Verify if $ABCD$ is a trapezoid and if it is isosceles.** Vertices: $A(-2,5)$, $B(-3,1)$, $C(6,1)$, $D(3,5)$ 1. Calculate slopes of sides: $$m_{AB} = \frac{1-5}{-3+2} = \frac{-4}{-1} = 4$$ $$m_{BC} = \frac{1-1}{6+3} = \frac{0}{9} = 0$$ $$m_{CD} = \frac{5-1}{3-6} = \frac{4}{-3} = -\frac{4}{3}$$ $$m_{DA} = \frac{5-5}{-2-3} = \frac{0}{-5} = 0$$ 2. Check for parallel sides: $BC$ and $DA$ both have slope $0$, so $BC \parallel DA$. 3. Since one pair of opposite sides is parallel, $ABCD$ is a trapezoid. 4. Check if trapezoid is isosceles by comparing lengths of non-parallel sides $AB$ and $CD$: $$AB = \sqrt{(-3+2)^2 + (1-5)^2} = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17}$$ $$CD = \sqrt{(3-6)^2 + (5-1)^2} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ Since $AB \neq CD$, trapezoid is not isosceles. **Answer:** $ABCD$ is a trapezoid but not isosceles. --- **Step 6: Solve Problem 13: Verify if $JKLM$ is a trapezoid and if it is isosceles.** Vertices: $J(-4,-6)$, $K(6,2)$, $L(1,3)$, $M(-4,-1)$ 1. Calculate slopes: $$m_{JK} = \frac{2+6}{6+4} = \frac{8}{10} = 0.8$$ $$m_{KL} = \frac{3-2}{1-6} = \frac{1}{-5} = -0.2$$ $$m_{LM} = \frac{-1-3}{-4-1} = \frac{-4}{-5} = 0.8$$ $$m_{MJ} = \frac{-1+6}{-4+4} = \frac{5}{0} = \text{undefined}$$ 2. Check for parallel sides: $JK$ and $LM$ both have slope $0.8$, so $JK \parallel LM$. 3. Since one pair of opposite sides is parallel, $JKLM$ is a trapezoid. 4. Check if trapezoid is isosceles by comparing lengths of non-parallel sides $KL$ and $MJ$: $$KL = \sqrt{(1-6)^2 + (3-2)^2} = \sqrt{(-5)^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26}$$ $$MJ = \sqrt{(-4+4)^2 + (-1+6)^2} = \sqrt{0 + 5^2} = 5$$ Since $KL \neq MJ$, trapezoid is not isosceles. **Answer:** $JKLM$ is a trapezoid but not isosceles. --- **Step 7: Solve Problem 14: Verify if $QRST$ is a trapezoid and if it is isosceles.** Vertices: $Q(2,5)$, $R(-2,1)$, $S(-1,-6)$, $T(9,4)$ 1. Calculate slopes: $$m_{QR} = \frac{1-5}{-2-2} = \frac{-4}{-4} = 1$$ $$m_{RS} = \frac{-6-1}{-1+2} = \frac{-7}{1} = -7$$ $$m_{ST} = \frac{4+6}{9+1} = \frac{10}{10} = 1$$ $$m_{TQ} = \frac{5-4}{2-9} = \frac{1}{-7} = -\frac{1}{7}$$ 2. Check for parallel sides: $QR$ and $ST$ both have slope $1$, so $QR \parallel ST$. 3. Since one pair of opposite sides is parallel, $QRST$ is a trapezoid. 4. Check if trapezoid is isosceles by comparing lengths of non-parallel sides $RS$ and $TQ$: $$RS = \sqrt{(-1+2)^2 + (-6-1)^2} = \sqrt{1^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50}$$ $$TQ = \sqrt{(2-9)^2 + (5-4)^2} = \sqrt{(-7)^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50}$$ Since $RS = TQ$, trapezoid is isosceles. **Answer:** $QRST$ is an isosceles trapezoid. --- **Step 8: Solve Problem 15: Verify if $WXYZ$ is a trapezoid and if it is isosceles.** Vertices: $W(-5,-1)$, $X(-2,2)$, $Y(3,1)$, $Z(5,-3)$ 1. Calculate slopes: $$m_{WX} = \frac{2+1}{-2+5} = \frac{3}{3} = 1$$ $$m_{XY} = \frac{1-2}{3+2} = \frac{-1}{5} = -0.2$$ $$m_{YZ} = \frac{-3-1}{5-3} = \frac{-4}{2} = -2$$ $$m_{ZW} = \frac{-3+1}{5+5} = \frac{-2}{10} = -0.2$$ 2. Check for parallel sides: $XY$ and $ZW$ both have slope $-0.2$, so $XY \parallel ZW$. 3. Since one pair of opposite sides is parallel, $WXYZ$ is a trapezoid. 4. Check if trapezoid is isosceles by comparing lengths of non-parallel sides $WX$ and $YZ$: $$WX = \sqrt{(-2+5)^2 + (2+1)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}$$ $$YZ = \sqrt{(5-3)^2 + (-3-1)^2} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20}$$ Since $WX \neq YZ$, trapezoid is not isosceles. **Answer:** $WXYZ$ is a trapezoid but not isosceles.