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 geometry problems: 1. Define parallel lines (1 mark) Parallel lines are lines in a plane that are always the same distance apart and never intersect or touch each other, no matter how far they are extended. 2. State the relationship between alternate interior angles when two parallel lines are cut by a transversal line (2 marks) When two parallel lines are cut by a transversal line, the alternate interior angles are equal in measure. 3. Find the value of x (6 marks) a) The angles x^ and 65^ are alternate interior angles. When two parallel lines are cut by a transversal, alternate interior angles are equal. x^ = 65^ x = 65 b) The angles 3x^ and (2x+10)^ are alternate interior angles. When two parallel lines are cut by a transversal, alternate interior angles are equal. Step 1: Set the angles equal to each other. 3x = 2x + 10 Step 2: Subtract 2x from both sides. 3x - 2x = 10 x = 10 Step 3: Verify the angles. 3x = 3(10) = 30^ 2x+10 = 2(10)+10 = 20+10 = 30^ x = 10 c) The angles (2x+15)^ and (3x-25)^ are consecutive interior angles (also known as same-side interior angles). When two parallel lines are cut by a transversal, consecutive interior angles are supplementary, meaning their sum is 180^. Step 1: Set the sum of the angles equal to 180^. (2x+15) + (3x-25) = 180 Step 2: Combine like terms. 5x - 10 = 180 Step 3