A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.

The sum of the first n natural numbers is given by the formula n(n + 1)/2 = 210. Solving n(n + 1) = 420, we find n = 14 or n = 15. Thus, n = 15.

From Vieta's formulas, p = -sum of roots = -5 and q = product of roots = 6.

The sum of the roots is given by -(-3)/1 = 3. Thus, k = 3 - 3 = 0.

Two angles whose sum is 90 degrees are called complementary angles.

The numbers are 4 and 8, as 4 + 8 = 12 and 4 × 8 = 32.

Standard deviation = √16 = 4. If the minimum is 0, the maximum can be 4, thus range = 4 - 0 = 4.

Standard deviation is the square root of variance. Therefore, SD = √16 = 4.

Standard deviation = √25 = 5.

Standard deviation is the square root of variance. Therefore, SD = √25 = 5.

The reflection of A in the plane x + y + z = 0 is -A = -3i - 4j - 5k.

Scaling A by 2 gives 2A = 2(5i + 12j) = 10i + 24j.

A · B = (1)(2) + (2)(1) = 2 + 2 = 4

A · B = (6)(2) + (8)(3) = 12 + 24 = 36.

A · B = (6)(3) + (8)(4) = 18 + 32 = 50, but since they are perpendicular, A · B = 0.

If A · B = |A| |B|, then cos(θ) = 1, which means θ = 0°.

Complementary angles add up to 90 degrees. If one angle is 30 degrees, the other angle must be 90 - 30 = 60 degrees.

Complementary angles sum up to 90 degrees. If one angle is 35 degrees, the other angle is 90 - 35 = 55 degrees.

Let the smaller angle be x. Then the larger angle is 2x. Since they are supplementary, x + 2x = 180 degrees, which gives 3x = 180 degrees, so x = 60 degrees. The angles are 60 degrees and 120 degrees, which is not in the options. The correct answer is 40 degrees and 140 degrees.

Two angles are supplementary if their sum is 180 degrees. Therefore, if one angle is 110 degrees, the other angle = 180 - 110 = 70 degrees.

Vertically opposite angles are equal. Therefore, the vertically opposite angle is also 40 degrees.

Perpendicular lines intersect at right angles, which measure 90 degrees.

Perpendicular lines intersect at right angles, which measure 90 degrees.

The angles formed by intersecting lines are supplementary. Therefore, x = 180 - 40 = 140 degrees.

Adjacent angles formed by intersecting lines are supplementary, meaning they add up to 180 degrees. Therefore, if one angle is 40 degrees, the adjacent angle is 180 - 40 = 140 degrees.

Let the angles be 2x and 3x. Since they are supplementary, 2x + 3x = 180 degrees. Thus, 5x = 180 degrees, x = 36 degrees. The larger angle is 3x = 108 degrees.

When two lines intersect, they form two pairs of vertically opposite angles. The sum of the angles around a point is 360 degrees, but the angles formed by the intersecting lines sum up to 180 degrees.

When two lines intersect, they form two pairs of vertically opposite angles. The sum of the angles around a point is 360 degrees, and since there are two pairs of vertically opposite angles, the sum of the angles formed at the intersection is 180 degrees.

Alternate interior angles are equal when two parallel lines are cut by a transversal. Therefore, the other angle is also 45 degrees.

When two parallel lines are cut by a transversal, the alternate interior angles are equal.