This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the questions. Question 1.1: To prove that the opposite sides of a parallelogram are equal, we can use congruent triangles. Consider parallelogram PQRS with diagonal PR. Step 1: Identify parallel lines and transversals. Since PQRS is a parallelogram, we know that PQ SR and PS QR. PR is a transversal for both pairs of parallel lines. Step 2: Identify alternate interior angles. Because PQ SR and PR is a transversal, QPR = SRP (alternate interior angles). Because PS QR and PR is a transversal, SPR = QRP (alternate interior angles). Step 3: Prove triangle congruence. Consider PQR and RSP: • QPR = SRP (from Step 2) • PR = RP (common side) • QRP = SPR (from Step 2) Therefore, PQR RSP (Angle-Side-Angle congruence criterion). Step 4: Conclude equality of opposite sides. Since PQR 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, G = 9x + 12^ and F = 15x - 24^. Question 1.2.1: Calculate the value of x. Step 1: Use the property of consecutive angles in a parallelogram. Consecutive angles in a parallelogram are supplementary, meaning they add up to 180^. G + F = 180^ Step 2: Substitute the given expressions for G and F and solve for x. (9x + 12^) + (15x - 24^) = 180^ 24x - 12^ = 180^ 24x = 180^ + 12^ 24x = 192^ x = (192)/(24) x = 8 Question 1.2.2: Determine the size of E. Step 1: Calculate the measure of G using the value of x. G = 9x + 12^ Substitute x=8: G = 9(8) + 12^ G = 72^ + 12^ G = 84^ Step 2: Use the property of opposite angles in a parallelogram. Opposite angles in a parallelogram are equal. Therefore, E = G. E = 84^ 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. Send me the next one 📸