Prove that opposite sides of a parallelogram are equal. Also, given parallelogram EFGH with angles G = 9x + 12 and F = 15x − 24, calculate the value of x.

Since PQRS is a parallelogram, we know that $PQ \parallel SR$ and $PS \parallel QR$.
PR is a transversal for both pairs of parallel lines.

Because $PQ \parallel SR$ and PR is a transversal, $\angle QPR = \angle SRP$ (alternate interior angles).
Because $PS \parallel QR$ and PR is a transversal, $\angle SPR = \angle QRP$ (alternate interior angles).

Consider $\triangle PQR$ and $\triangle RSP$:

• $\angle QPR = \angle SRP$ (from Step 2)
• $PR = RP$ (common side)
• $\angle QRP = \angle SPR$ (from Step 2) Therefore, $\triangle PQR \cong \triangle RSP$ (Angle-Side-Angle congruence criterion).

Since $\triangle PQR \cong \triangle RSP$, their corresponding sides are equal:
$PQ = RS$
$QR = SP$
Thus, the opposite sides of a parallelogram are equal.

Question 1.2:
In parallelogram EFGH, $\hat{G} = 9x + 12^\circ$ and $\hat{F} = 15x - 24^\circ$.

Question 1.2.1:
Calculate the value of $x$.

Consecutive angles in a parallelogram are supplementary, meaning they add up to $180^\circ$.

Question 1.2.2:
Determine the size of $\hat{E}$.

Substitute $x=8$:

Opposite angles in a parallelogram are equal. Therefore, $\hat{E} = \hat{G}$.

Question 2:
The question provides the vertices of a quadrilateral NTWZ as N (-6;-6), T (3;0), W (x;y) and Z (5;-3). However, the actual question to be solved (e.g., "find x and y if NTWZ is a parallelogram", or "calculate the length of a side") is not visible in the image. Please provide the complete question for Question 2.