D

The total number of degrees in a hexagon is 720; if you don’t know the formula, (s – 2) × 180, you can divide the hexagon into 4 non-overlapping triangles. Subtract the two known angles, leaving you with 508° for the four remaining angles. Since the remaining angles are equal, each angle is 508 ÷ 4 = 127°.

E

and F Because they are supplementary angles, a + b = 180. So subtract the range of values for a from 180 to get 116 < b < 150. You know that b and d are equal, so double b to get 232 < b + d < 300. Only choices (E) and (F) fall within this range. (You could also Plug In the Answers on this question.)

30

If you draw a line down from B to the base of the figure, you can subtract the 90° that are left over from the 150° angle and you’ll have 60° left.This makes a 30-60-90 triangle with vertex D. And if ∆CED is isosceles, that makes angle CED 30° as well.

180

You don’t actually have to do any math for this question. When parallel lines intersect, any big angle plus any little angle is 180°; since x is a little angle and y is a big angle, the sum must be 180. However, you could also use the rules regarding opposite and corresponding angles, or the parallelogram rules, with the 75° in the corner. In this case, x = 75 and y = 105, so 75 + 105 = 180.

C, D, and G

Remember that when a line intersects two parallel lines, it makes large and small angles; all of the large angles are equal, as are all of the small ones. In this case, s is equal to the other large angle measures: v, w, and z.Choices (C), (D), and (G) work.

Use the laws of parallel lines to fill in the diagram. ∠ABD + ∠BDE + ∠CDE = 180° because lines AB and CD are parallel. ∠CDE = 180° – x° – 44°.Therefore, 3x + 2x + 180 – x – 44 = 180. Solving for x gives you 11, and ∠ABD = 33°.

15

The formula for the total interior angles of a polygon with n sides is (n – 2)180, so the interior angles of an 8-sided polygon total 6 × 180 = 1080°. Since it’s a regular polygon, divide that total by the 8 angles to determine that p = 135 when n = 8. For the 6-sided polygon, the total of the interior angles is 4 × 180 = 720°, and each angle is 720 ÷ 6 = 120.Thus p = 120 when n = 6, and 135 – 120 = 115.

Plug in values for the unknown angles. When a = 60 and b = 130, the angle vertical to a also measures 60°, and the angle adjacent to b within the triangle must measure 180° – 130° = 50°.The sum of the angles in a triangle is 180°.Therefore, the remaining angle measures 180° – 60° – 50° = 70°. Angle c is vertical to the 70° angle, so c = 70. Quantity A is 60 + 70 = 130 and Quantity B is 130; the quantities are equal.Eliminate choices (A) and (B). Plugging in a second set of numbers will show you that any set of numbers yields the same result, so the answer is choice (C). Alternatively, you could use algebra to determine that the three angles in the triangle measure a°, (180 – b)°, and c°.Therefore, a + (180 – b) + c = 180. Subtract 180 from each side of this equation and add b to each side; a + c = b.The quantities are equal.

The angle between the ones marked x° and 3x° is vertical to the one that measures 4y°.These three angles form a straight line, so x + 4y + 3x = 180. Since 4y = 5x, x + 5x + 3x = 180; 9x = 180; x = 20.Therefore 4y = 5x = 100; y = 25.

If f = 130, then both large angles formed by lines t and k also measure 130°.The small angles formed by those two lines therefore measure 50° (notice that one of these angles is the left base angle of the triangle). If g = 70, then the angle above it (the other base angle of the triangle) must measure 110° to complete the 180° in a straight line. So far, you have 160° in the triangle.To complete the 180° total in the triangle, h must measure 20°.The answer is choice (B).