Solve for the marked sides in the given triangles. Assume all triangles are right-angled.

Here are the solutions for the marked sides: a) Step 1: Identify the trigonometric relationship. In the right-angled triangle, we are given the angle 45^, the side opposite to it (80), and we need to find the hypotenuse (x). The sine function relates these: () = oppositehypotenuse Step 2: Substitute the values and solve for x. (45^) = (80)/(x) x = (80)/((45^)) Since (45^) = sqrt(2)2: x = (80)/(sqrt(2))2 = (160)/(sqrt(2)) = 160sqrt(2)2 = 80sqrt(2) x ≈ 80 × 1.4142 = 113.136 The value of x is 80sqrt(2). b) This problem involves two right-angled triangles sharing a common altitude. Let h be the common altitude. Step 1: Calculate the altitude h using the left triangle. In the left right-angled triangle, the hypotenuse is 8 cm and the angle is 60^. (60^) = (h)/(8 cm) h = 8 cm × (60^) = 8 cm × sqrt(3)2 = 4sqrt(3) cm Step 2: Calculate the hypotenuse Hy of the right triangle. In the right right-angled triangle, the altitude is h = 4sqrt(3) cm and the angle is 45^. (45^) = (h)/(Hy) Hy = (h)/((45^)) = 4sqrt(3) cmsqrt(2)2 = 8sqrt(3) cmsqrt(2) = 8sqrt(3)sqrt(2) cm2 = 4sqrt(6) cm Step 3: Calculate the total base Y. First, find the base segment of the left triangle (x_1) and the right triangle (x_2). For the left triangle: (60^) = (x_1)/(8 cm) x_1 = 8 cm × (60^) = 8 cm × (1)/(2) = 4 cm For the right triangle: (45^) = (h)/(x_2) x_2 = (h)/((45^)) = 4sqrt(3) cm1 = 4sqrt(3) cm The total base Y is the sum of x_1 and x_2: Y = x_1 + x_2 = 4 cm + 4sqrt(3) cm = 4(1+sqrt(3)) cm The marked sides are Hy = 4sqrt(6) cm and Y = 4(1+sqrt(3)) cm. c) This diagram shows a right-angled triangle with an angle of 60^. The hypotenuse is marked x and the side adjacent to the $