Prove that the opposite sides of a parallelogram are equal in length.

Fresh day olgathobakgale89, let's solve. Question 1 1.1 To prove that the opposite sides of parallelogram PQRS are equal, we can use the diagonal QS to divide the parallelogram into two triangles, PQS and RSQ. Step 1: Identify parallel lines and transversals. Since PQRS is a parallelogram, PQ RS and PS QR. QS is a transversal for both pairs of parallel lines. Step 2: Identify equal alternate interior angles. Because PQ RS and QS is a transversal, the alternate interior angles are equal: PQS = RSQ (or Q_1 = S_2 from the diagram) Because PS QR and QS is a transversal, the alternate interior angles are equal: PSQ = RQS (or S_1 = Q_2 from the diagram) Step 3: Identify the common side. The side QS is common to both PQS and RSQ. Step 4: Conclude triangle congruence. By the Angle-Side-Angle (ASA) congruence criterion, since two angles and the included side of PQS are equal to two angles and the included side of RSQ: PQS RSQ Step 5: Conclude equality of opposite sides. Since the triangles are congruent, their corresponding sides are equal: PQ = RS PS = QR Therefore, the opposite sides of the parallelogram PQRS are equal. 1.2 Assuming the parallelogram in the diagram is EFGH, as shown, and the given angles G and F refer to this parallelogram. 1.2.1 Calculate the value of x. Step 1: In a parallelogram, consecutive angles are supplementary (they add up to 180^). Therefore, G + F = 180^. Step 2: Substitute the given expressions for G and F into the equation. (9x + 12^) + (15x - 24^) = 180^ Step 3: Solve for x. 24x - 12^ = 180^ 24x = 180^ + 12^ 24x = 192^ x = (192)/(24) x = 8 1.2.2 Determine the size of E. Step 1: In a parallelogram, opposite angles are equal. Therefore, E = G. Step 2: Substitute the value of x into the expression for G. G = 9x + 12^ G = 9(8) + 12^ G = 72^ + 12^ G = 84^ Step 3: Since E = G, the size of E is: E = 84^ Send me the next one 📸