Let's break down and solve the problems step by step:

Question 22

Vertices: $W(-2, 4), X(5, 5), Y(6, -2), Z(-1, -3)$

Use the slope formula:
$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

1. Find slopes of opposite sides:

   • Slope of $WX$:
     $\frac{5 - 4}{5 - (-2)} = \frac{1}{7}$

   • Slope of $YZ$:
     $\frac{-3 - (-2)}{-1 - 6} = \frac{-1}{-7} = \frac{1}{7}$

   • Slope of $XY$:
     $\frac{-2 - 5}{6 - 5} = \frac{-7}{1} = -7$

   • Slope of $WZ$:
     $\frac{-3 - 4}{-1 - (-2)} = \frac{-7}{1} = -7$

   Since opposite sides have equal slopes ($WX \parallel YZ$ and $WZ \parallel XY$), $WXYZ$ is a parallelogram.
   Check perpendicularity (rectangle): Since adjacent sides have negative reciprocal slopes ($WX = \frac{1}{7}$ and $WZ = -7$), it's a rectangle.

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Question 23

Vertices: $J(3, 3), K(-5, 2), L(-4, -4), M(4, -3)$

Use the distance formula:
$\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

1. Find lengths of opposite sides:

   • Length of $JK$:
     $\sqrt{(-5 - 3)^2 + (2 - 3)^2} = \sqrt{(-8)^2 + (1)^2} = \sqrt{64 + 1} = \sqrt{65}$

   • Length of $LM$:
     $\sqrt{(4 - (-4))^2 + (-3 - (-4))^2} = \sqrt{(8)^2 + (1)^2} = \sqrt{65}$

   • Length of $KL$:
     $\sqrt{(-4 - (-5))^2 + (-4 - 2)^2} = \sqrt{(1)^2 + (-6)^2} = \sqrt{1 + 36} = \sqrt{37}$

   • Length of $JM$:
     $\sqrt{(4 - 3)^2 + (-3 - 3)^2} = \sqrt{(1)^2 + (-6)^2} = \sqrt{37}$

   Opposite sides are equal, so $JKLM$ is a parallelogram.

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Question 24

Vertices: $Q(-2, 2), R(0, -2), S(6, 1), T(4, 5)$

1. Use the distance formula to verify if diagonals bisect each other (for parallelogram property).

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Questions 26-31

Given $m \angle Z = 40^\circ$:

• $\angle 1$: $90^\circ$
• $\angle 7$: $50^\circ$
• $\angle 3$: $90^\circ$
• $\angle 5$: $90^\circ$.

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Let me know which part needs further clarification or elaboration!

Related Questions to Explore:

1. What makes a parallelogram a rectangle based on slopes?
2. How do you calculate slopes if one vertex is missing?
3. What is the difference between parallelograms and rhombi?
4. How can diagonals confirm the properties of quadrilaterals?
5. How does the sum of angles verify a rectangle?

Tip: Always check both slopes and distances when verifying rectangle properties.