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 for the geometry problems. Question 2 a) The diagram shows two parallel horizontal lines intersected by two transversals, forming an isosceles triangle. Step 1: Find the size of angle z. Angle z and the given 60^ angle are alternate interior angles because they are between parallel lines and on opposite sides of a transversal. Alternate interior angles are equal. z = 60^ The size of angle z is 60^ (Reason: Alternate interior angles). Step 2: Find the size of angle y. The triangle is isosceles, and angle z is one of its base angles. Therefore, the other base angle (at the top-left vertex of the triangle) is also 60^. The sum of angles in a triangle is 180^. y + 60^ + 60^ = 180^ y + 120^ = 180^ y = 180^ - 120^ y = 60^ The size of angle y is 60^ (Reason: Sum of angles in an isosceles triangle). Step 3: Find the size of angle x. Angle x and angle y are vertically opposite angles. Vertically opposite angles are equal. x = y x = 60^ The size of angle x is 60^ (Reason: Vertically opposite angles). Step 4: Find the size of angle w. Angle w and the top-left base angle of the isosceles triangle (which is 60^) are alternate interior angles because they are between parallel lines and on opposite sides of a transversal. Alternate interior angles are equal. w = 60^ The size of angle w is 60^ (Reason: Alternate interior angles). b) In the diagram, PQ is parallel to RS, and PY = PZ. Step 1: Find the interior angle at R. The angle 140^ and the interior angle PRS form a linear pair on a straight line. Angles on a straight line sum to 180^. PRS = 180^ - 140^ PRS = 40^ Step 2: Find angle YPZ. Since PQ is parallel to RS, angle YPZ and angle PRS are alternate interior angles. Alternate interior angles are equal. YPZ = PRS YPZ = 40^ Step 3: Find the base angles of triangle PYZ. Given that PY = PZ, triangle PYZ is an isosceles triangle. The base angles opposite the equal sides are equal, so PYZ = PZY. The sum of angles in a triangle is 180^. YPZ + PYZ + PZY = 180^ 40^ + PZY + PZY = 180^ 40^ + 2 · PZY = 180^ Step 4: Solve for PZY, which is x. 2 · PZY = 180^ - 40^ 2 · PZY = 140^ PZY = (140^)/(2) PZY = 70^ Therefore, the size of angle x is 70^. Question 3 a) The diagram shows a trapezium with parallel top and bottom sides. The angles are given in terms of x. For parallel lines intersected by a transversal, consecutive interior angles sum to 180^. Step 1: Calculate x using the angles on the left transversal. The angles 2x (top left) and 2x + 10^ (bottom left) are consecutive interior angles. 2x + (2x + 10^) = 180^ 4x + 10^ = 180^ 4x = 180^ - 10^ 4x = 170^ x = (170^)/(4) x = 42.5^ Step 2: Calculate x using the angles on the right transversal. The angles 3x - 20^ (top right) and 2x (bottom right) are consecutive interior angles. (3x - 20^) + 2x = 180^ 5x - 20^ = 180^ 5x = 180^ + 20^ 5x = 200^ x = (200^)/(5) x = 40^ Step 3: Identify the inconsistency. The problem contains an inconsistency, as the value of x differs depending on which pair of consecutive interior angles is used (42.5^ from the left side and 40^ from the right side). Therefore, a unique value for x cannot be determined from the given information. Step 4: Provide a value with a note. Since a unique value for x cannot be determined due to the inconsistency in the problem statement, we will provide the value derived from the left side as an example. The value of x is 42.5^ (Note: The problem contains an inconsistency, leading to different values of x depending on the transversal used). That's 2 down. 3 left today — send the next one.