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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1.1 Use the diagram below to prove that the opposite sides of a parallelogram are equal. Step 1: Consider the parallelogram PQRS with diagonal PR. Given that PQRS is a parallelogram, we know that its opposite sides are parallel: PQ SR and PS QR. Step 2: Identify alternate interior angles. Since PQ SR and PR is a transversal, the alternate interior angles are equal: QPR = SRP Since PS QR and PR is a transversal, the alternate interior angles are equal: SPR = QRP Step 3: Identify the common side. The diagonal PR is common to both PQR and RSP. PR = RP Step 4: Prove triangle congruence. By the Angle-Side-Angle (ASA) congruence criterion, PQR is congruent to RSP: PQR RSP (ASA) Step 5: Conclude equality of opposite sides. Since the triangles are congruent, their corresponding sides are equal: PQ = RS QR = SP Therefore, the opposite sides of a parallelogram are equal. 1.2 In the diagram below ABCD is a parallelogram with G = 9x + 12^ and F = 15x - 24^. (Assuming the parallelogram in the diagram is EFGH, and the given angles G and F refer to G and F of parallelogram EFGH.) 1.2.1 Calculate the value of x. Step 1: Identify the property of consecutive angles in a parallelogram. In a parallelogram, consecutive angles are supplementary, meaning they add up to 180^. Therefore, G + F = 180^. Step 2: Set up the equation. Substitute the given expressions for G and F: (9x + 12^) + (15x - 24^) = 180^ Step 3: Solve for x. Combine like terms: 24x - 12^ = 180^ Add 12^ to both sides: 24x = 180^ + 12^ 24x = 192^ Divide by 24: x = (192)/(24) x = 8 The value of x is 8. 1.2.2 Determine the size of E. Step 1: Identify the property of opposite angles in a parallelogram. In a parallelogram, opposite angles are equal. Therefore, E = G. Step 2: Calculate the measure of G using the value of x. Substitute x = 8 into the expression for G: G = 9x + 12^ G = 9(8) + 12^ G = 72^ + 12^ G = 84^ Step 3: Determine the size of E. Since E = G: E = 84^ The size of E is 84^. Send me the next one 📸