Alternate exterior angles are equal, so angle B also measures 120 degrees.

Alternate exterior angles are equal, so angle B also measures 45 degrees.

Alternate interior angles are equal, so angle B also measures 75 degrees.

Corresponding angles are equal when two parallel lines are cut by a transversal.

Same-side interior angles are supplementary when two parallel lines are cut by a transversal.

Same-side interior angles are supplementary, so angle B = 180 - 75 = 105 degrees.

Same-side interior angles are supplementary. Therefore, angle B = 180 - 65 = 115 degrees.

Since angle C and angle D are interior angles on the same side of the transversal, they are supplementary. Therefore, angle D = 180 - 30 = 150 degrees.

Corresponding angles are equal when a transversal intersects two parallel lines. Therefore, the other corresponding angle also measures 30 degrees.

Corresponding angles are equal, so angle C also measures 85 degrees.

Complementary angles sum to 90°. Therefore, angle Y = 90° - 35° = 55°.

Using sin²θ + cos²θ = 1, sin²θ = 1 - 0.6² = 0.64, so sin(θ) = 0.8.

θ = arccos(0.8) ≈ 36.87°, rounded to 37°

Substituting x = 3 into the function: f(3) = 2(3)^2 - 8(3) + 6 = 18 - 24 + 6 = 0.

Substitute x = 2 into the function: f(2) = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3.

The polynomial can be factored as (x - 2)(x + 2). Setting each factor to zero gives us x - 2 = 0 or x + 2 = 0, so the roots are x = -2 and x = 2.

Substituting x = 2 into the function gives f(2) = 2^2 - 4(2) + 4 = 0.

Factoring gives f(x) = (x - 2)(x - 4). Setting each factor to zero gives the roots x = 2 and x = 4.

Substituting x = 3 into the function: f(3) = 3^2 - 6(3) + 9 = 9 - 18 + 9 = 0.

We can factor the polynomial as f(x) = (x - 3)(x + 3). Setting each factor to zero gives us the roots x = -3 and x = 3.

The slopes of perpendicular lines are negative reciprocals. The negative reciprocal of 4 is -1/4.

The third angle can be found by subtracting the sum of the other two angles from 180 degrees: 180 - (50 + 60) = 70 degrees.

In a triangle, the sum of the angles is 180 degrees. If one angle is 90 degrees, the remaining two angles must sum to 90 degrees. Since they are equal, each must be 45 degrees.

If one root is 2, then the other root can be found using the product of the roots: 2 * r = 6, so r = 3. The sum of the roots is 2 + 3 = -p, thus p = -5.

Using the sum of the roots, p = -(3 + 1) = -4. The product of the roots gives q = 3 * 1 = 3.

Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) = ((1*4 + 2*1)/(1+2), (1*6 + 2*2)/(1+2)) = (3, 4).

Using the section formula: C = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) where m=1, n=2. C = ((1*4 + 2*1)/(1+2), (1*6 + 2*2)/(1+2)) = (3, 4).

Using the section formula: C = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) where m=1, n=2. C = ((1*4 + 2*1)/(1+2), (1*6 + 2*2)/(1+2)) = (2.67, 4).

Using the section formula: C = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) = ((1*4 + 2*1)/(1+2), (1*6 + 2*2)/(1+2)) = (3, 5).

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).