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: Problem 1 (Top left triangle, labeled "1.2") Step 1: The sum of angles in a triangle is 180^. (3x - 15^) + 62^ + 34^ = 180^ 3x - 15 + 96 = 180 3x + 81 = 180 Step 2: Solve for x. 3x = 180 - 81 3x = 99 x = (99)/(3) x = 33 Problem 2 (Top right triangle) Step 1: Angles on a straight line sum to 180^. The interior angle x and the exterior angle 130^ form a linear pair. x + 130^ = 180^ x = 180^ - 130^ x = 50^ Step 2: The exterior angle of a triangle is equal to the sum of the two opposite interior angles. 130^ = (y - 20^) + y 130 = 2y - 20 Step 3: Solve for y. 130 + 20 = 2y 150 = 2y y = (150)/(2) y = 75^ Problem 3 (Middle left figure) Step 1: Lines AB and DC are parallel (indicated by arrows). BC is a transversal. Consecutive interior angles sum to 180^. ABC + BCD = 180^ y + 90^ = 180^ y = 180^ - 90^ y = 90^ Step 2: In BCE, the sum of angles is 180^. BEC = 180^ - BCE - CBE BEC = 180^ - 90^ - 40^ BEC = 50^ Step 3: Assuming the arrow on DE indicates that DE is parallel to BC. If DE || BC, then DEC and BCE are consecutive interior angles, so they sum to 180^. x + 90^ = 180^ x = 180^ - 90^ x = 90^ Problem 4 (Middle right, labeled "10.3") Step 1: The two horizontal lines are parallel. The angles 6c - 40^ and 2c - 20^ are consecutive interior angles, so they sum to 180^. (6c - 40^) + (2c - 20^) = 180^ 8c - 60 = 180 Step 2: Solve for c. 8c = 180 + 60 8c = 240 c = (240)/(8) c = 30 Step 3: Angle d and the angle 6c - 40^ are vertically opposite angles, so they are equal. d = 6c - 40^ d = 6(30) - 40 d = 180 - 40 d = 140^ Problem 5 (Bottom left, intersecting lines) Step 1: Vertically opposite angles are equal. t + 30^ = 60^ Step 2: Solve for t. t = 60 - 30 t = 30 Problem 6 (Bottom left triangle) Step 1: The markings on the sides indicate that all three sides of the triangle are equal, meaning it is an equilateral triangle. All angles in an equilateral triangle are 60^. 2x - 30^ = 60^ Step 2: Solve for x. 2x = 60 + 30 2x = 90 x = (90)/(2) x = 45 Problem 7 (Bottom middle, angles on a straight line) Step 1: Angles on a straight line sum to 180^. The square symbol indicates a 90^ angle. 20^ + 90^ + a = 180^ 110^ + a = 180^ Step 2: Solve for a. a = 180^ - 110^ a = 70^ Problem 8 (Bottom right triangle, labeled "10.2") Step 1: The interior angle adjacent to the 130^ exterior angle forms a linear pair, so it is 180^ - 130^ = 50^. This is the vertex angle of the isosceles triangle. Step 2: The markings indicate that the triangle is isosceles, and the two angles marked b are the base angles, which are equal. The sum of angles in a triangle is 180^. b + b + 50^ = 180^ 2b + 50 = 180 Step 3: Solve for b. 2b = 180 - 50 2b = 130 b = (130)/(2) b = 65^ Send me the next one 📸