Solutions

1. 95°

   You know that x + y = 180, because any acute angle formed by a transversal that cuts across two parallel lines is supplementary to any obtuse angle formed in the same figure. Use the information given to set up a system of two equations with two variables:

   
2. 72°

   Set up a ratio, using the unknown multiplier, a:

   Set x = 3a and y = 2a, so
   180 = x + y = 3a + 2a = 5a
   180 = 5a
   a = 36
   y = 2a = 2(36) = 72

3. 140°

   Use the fact that x + y = 180 to set up a system of two equations with two variables:

   
4. 120°

   First, simplify the given equation. Then, use the fact that x + y = 180 to set up a system of two equations with two variables.
5. 95°

   Because a and d are vertical angles, they have the same measure: a = d = 95. Likewise, because b and e are vertical angles, they have the same measure: b = e. Therefore, b + d − e = d = 95.

6. 65°

   Because c and f are vertical angles, they have the same measure: c + f = 70, so c = f = 35. Notice that b, c, and d form a straight line: b + c + d = 180. Substitute the known values of c and d into this equation:

   

7. b and d, a and e, & d and e

   If a is complementary to b, then d (which is equal to a, since they are vertical angles), is also complementary to b. Likewise, if a is complementary to b, then a is also complementary to e (which is equal to b, since they are vertical angles). Finally, d and e must be complementary, since d = a and e = b. You do not need to know the term “complementary,” but you should be able to work with the concept (two angles adding up to 90°).

8. 315°

   If e = 45, then the sum of all the other angles is 360 − 45 = 315.

9. 105°

   You are told that a + e = 150. Because they are both acute angles formed by a transversal cutting across two parallel lines, they are also equal. Therefore, a = e = 75. Any acute angle in this diagram is supplementary to any obtuse angle, so 75 + f = 180, and f = 105.

10. 70°

    You know that angles a and g are supplementary; their measures sum to 180. Therefore:

    Angle f is equal to angle g, so its measure is also 3y + 20. The measure of angle f = g = 3(40) + 20 = 140. If f = 2x, then 140 = 2x and, therefore x = 70.

11. 70°

    You are given the measure of one acute angle (a) and one obtuse angle (g). Because any acute angle in this diagram is supplementary to any obtuse angle, then 11y + 4x − y = 180, or 4x + 10y = 180. Because angle d is equal to angle a, then 5y + 2x − 20 = 4x − y, or 2x − 6y = −20. You can set up a system of two equations with two variables:

    

    h is one of the acute angles, therefore, h has the same measure as a: 4x − y = 4(20) − 10 = 70.

12. 68°

    Because b and d are supplementary, 4x + 3y + 8 = 180 or 4x + 3y = 172. Because d and e are equal, 3y + 8 = x + 2y or x − y = 8. You can set up a system of two equations with two variables:

    

    Because h is equal to e, h = x + 2y, or 28 + 2(20) + 8 = 68.

13. 40°

    If c + f = 140, then i = 40, because there are 180° in a triangle. Because k is vertical to i, k is also equal to 40. Alternatively, if c + f = 140, then l = 140, since l is an exterior angle of the triangle and is therefore equal to the sum of the two remote interior angles. Because k is supplementary to l, k = 180 − 140 = 40.

14. 90°

    If f = 90, then the other two angles in the triangle, c and i, sum to 90. a and k are vertical angles to c and i, therefore, they sum to 90 as well.

15. 150°

    Angle k is vertical to angle i. So if f + k = 150, then f + i = 150. Angle b, an exterior angle of the triangle, must be equal to the sum of the two remote interior angles, f and i. Therefore, b = 150.

16. (C)

    You can substitute each of the values in Quantity A for a corresponding value in Quantity B: a = d, c = f, and b = e, in each case because the equal angles are vertical angles. Rewrite Quantity A:

    Quantity A	Quantity B
    a + f + b = d + c + e	c + d + e

    Alternatively, you could have noted that the angles measured by a, f, and b together form a straight line. So a + f + b must be 180. Likewise, c + d + e must be 180, because the corresponding angles form the same straight line (from the other side). By either argument, the two quantities are equal.

17. (C)

    To see why the sums in the two quantities are equal, label the remaining two interior angles of the quadrilateral according to the rules for supplementary angles:

    

    Quantity A	Quantity B
    w + y	x + z

    There are several relationships that can be described based on the diagram. For instance, you know the sum of the four internal angles of the quadrilateral is 360:

    

    Therefore, the two quantities are equal.

18. (A)

    First solve for x. The two angles x and 2x are supplementary:

    

    Next note that 2x = 6y, because 2x and 6y are vertical angles. Plug in 60 for x and solve for y:

    

    Quantity A	Quantity B
    y = 20	10

    Therefore, Quantity A is greater.