Geometry Practice Problems for GED Math (page 2)

Answers

1. d.   Angles 2, 4, 6, and 7 are alternating (vertical) angles. Therefore, their measures are equal. Angles 7 and 8 are supplementary. Therefore, angles 2 and 8 are also supplementary. 12x + 10 + 7x – 1 = 180, 19x + 9 = 180, 19x = 171, x = 9. Because x = 9, the measure of angle 2 is 12(9) + 10 = 108 + 10 = 118 degrees.
2. e.   Angles 5 and 6 are supplementary. Therefore, the sum of their measures is 180 degrees. If the measure of angle 6 is x, then the measure of angle 5 is 5x. 5x + x = 180, 6x = 180, x = 30. The measure of angle 6 is 30 degrees, and the measure of angle 5 is 5(30) = 150 degrees.
3. c.   Angles 4 and 7 are alternating angles. Therefore, their measures are equal: 6x + 20 = 10x – 40, 4x = 60, x = 15. Because x = 15, the measure of angles 4 and 7 is 6(15) + 20 = 90 + 20 = 110. Notice that replacing x with 15 in the measure of angle 7 also yields 110: 10(15) – 40 = 150 – 40 = 110. Because angles 6 and 7 are vertical angles, the measure of angle 6 is also 110 degrees.
4. a.   If angle 3 measures 90 degrees, then angles 1, 6, and 7 must also measure 90 degrees, because they are alternating angles. Angles 3 and 4 are supplementary, because these angles form a line. Therefore, the measure of angle 4 is equal to 180 – 90 = 90 degrees. Angles 3 and 4 are congruent and supplementary. Because angles 2, 4, 5, and 8 are alternating angles, they are all congruent to each other. Every numbered angle measures 90 degrees. Therefore, every numbered angle is congruent and supplementary to every other numbered angle. Angles 5 and 7 are in fact adjacent, because they share a common vertex and a common ray. However, angles 1 and 2 are not complementary—their measures add to 180 degrees, not 90 degrees.
5. b.   Angles 2 and 6 are alternating angles. Therefore, their measures are equal. 8x + 10 = x² – 38, x² – 8x – 48 = 0. Factor x² – 8x – 48 and set each factor equal to 0. x² – 8x – 48 = (x + 4)(x – 12), x + 4 = 0, x = –4. x – 12 = 0, x = 12. An angle cannot have a negative measure, so the –4 value of x must be discarded. If x = 12, then the measure of angle 2 is 8(12) + 10 = 106 degrees. Notice that replacing x with 12 in the measure of angle 6 also yields 106: (12)² – 38 = 144 – 38 = 106. Angles 6 and 8 are supplementary, so the measure of angle 8 is equal to 180 – 106 = 74 degrees.
6. e.   If x is the width of the rectangle, then 2x – 4 is the length of the rectangle. Because opposite sides of a rectangle are congruent, the perimeter of the rectangle is equal to 2x – 4 + x + 2x – 4 + x = 6x – 8.
7. c.   One face of a cube is a square. The area of a square is equal to the length of one side of the square multiplied by itself. Therefore, the length of a side of this square (and edge of the cube) is equal to or units. Because every edge of a cube is equal in length and the volume of a cube is equal to e³, where e is the length of an edge (or lwh, l is the length of the cube, w is the width of the cube, and h is the height of the cube, which in this case, are all units), the volume of the cube is equal to cubic units.
8. d.   The circumference of a circle is equal to 2πr. Because the radius of a circle is half the diameter of a circle, 2r is equal to the diameter, d, of a circle. Therefore, the circumference of a circle is equal to πd. If the diameter is doubled, the circumference becomes π2d, or two times its original size.
9. b.   If the base of the triangle is b, then the height of the triangle is . The area of a triangle is equal to . Therefore, the area of this triangle is equal to .
10. e.   Use the Pythagorean theorem: (x – 3)² + (x + 4)² = (2x – 3)², x² – 6x + 9 + x² + 8x + 16 = 4x² – 12x + 9, 2x² + 2x + 25 = 4x² – 12x + 9, 2x² – 14x – 16 = 0, x² – 7x – 8 = 0, (x – 8)(x + 1) = 0, x = 8. Disregard the negative value of x, because a side of a triangle cannot be negative. Because x = 8, the length of the hypotenuse is 2(8) – 3 = 16 – 3 = 13.
11. b.   The slope of a line is the difference between the y values of two points divided by the difference between the x values of those two points: (4 – 6) ÷ 7 – (–3) = .
12. b.   The midpoint of a line segment is equal to the average of the x values of the endpoints and the average of the y values of the endpoints:



13. c.   To find the distance between two points, use the distance formula:

14. a.   Perpendicular lines cross at right angles. Therefore, angle AOC is 90 degrees. Because angles 1 and 2 combine to form angle AOC, the sum of the measures of angles 1 and 2 must be 90 degrees. Therefore, they are complementary angles.
15. d.   AE is perpendicular to OC, so angles AOC and EOC are both 90 degrees. Because angles 1 and 2 combine to form angle AOC and angles 3 and 4 combine to form angle EOC, these sums must both equal 90 degrees. Therefore, angle 1 + angle 2 = angle 3 + angle 4. Angles 1, 2, 3, and 7 form a line, as do angles 4, 5, and 6. Therefore, the measures of angles 1, 2, 3, and 7 add to 180 degrees, as do the measures of angles 4, 5, and 6. In the same way, angles 2, 3, 4, and 5 form a line, so the sum of the measures of those angles is also 180 degrees. Angles GOF and BOD are vertical angles. Therefore, their measures are equal. Angle 6 is angle GOF and angles 2 and 3 combine to form angle BOD, so the measure of angle 6 is equal to the sum of the measures of angles 2 and 3. However, the sum of angle 1 and angle 7 is not equal to the sum of angle 2 and angle 3. In fact, the sum of angles 2 and 3 is supplementary to the sum of angles 7 and 1, because angle 6 is supplementary to the sum of angles 7 and 1, and the sum of angles 2 and 3 is equal to the measure of angle 6.
16. c.   Because AE is perpendicular to OC, angle EOC is a right angle (measuring 90 degrees). Angles 3 and 4 combine to form angle EOC; therefore, their sum is equal to 90 degrees: 2x + 2 + 5x – 10 = 90, 7x – 8 = 90, 7x = 98, x = 14. Angle 4 is equal to 5(14) – 10 = 70 – 10 = 60 degrees. Because angles 4 and 7 are vertical angles, their measures are equal, and angle 7, too, measures 60 degrees.
17. b.   Because AE is perpendicular to OC, angle EOC is a right angle (measuring 90 degrees). Angles 3 and 4 combine to form angle EOC; therefore, their sum is equal to 90 degrees: 90 – 57 = 33. Angle 3 measures 33 degrees. Angle AOC is also a right angle, with angles 1 and 2 combining to form that angle. Therefore, the measure of angle 2 is equal to: 90 – 62 = 28 degrees. Angle 6 and angle BOD are vertical angles; their measures are equal. Because angles 2 and 3 combine to form angle BOD, the measure of BOD (and angle 6) is equal to: 33 + 28 = 61 degrees.
18. b.   Because AE is perpendicular to OC, angle EOC is a right angle (measuring 90 degrees). Angles 3 and 4 combine to form angle EOC; therefore, their sum is equal to 90 degrees: 5x + 3 + 15x + 7 = 90, 20x + 10 = 90, 20x = 80, x = 4. Therefore, the measure of angle 4 is equal to: 15(4) + 7 = 67 degrees. Angles 4, 5, and 6 form a line; therefore, the sum of their measures is 180 degrees. If x is the sum of angles 5 and 6, then 67 + x = 180, and x = 113 degrees.
19. a.   If the square and the rectangle share a side, then the width of the rectangle is equal to the length of the square. If the length of the square is x, then x + x + x + x = 2, 4x = 2, x = . Because the width of the rectangle is equal to the length of the square and the length of the rectangle is 4 times the length of the square, the width of the rectangle is and the length is 4() = 2. Therefore, the perimeter of the rectangle is equal to: + 2 + + 2 = 5 units.
20. c.   The area of a rectangle is equal to its length times its width. Therefore, the width of the rectangle is equal to its area divided by its length: . x² + 7x + 10 can be factored into (x + 2)(x + 5). Cancel the (x + 2) terms from the numerator and denominator of the fraction. The width of the rectangle is x + 5 units.
21. e.   The volume of a rectangular solid is equal to lwh, where l is the length of the solid, w is the width of the solid, and h is the height of the solid. If x represents the width (and therefore, the height as well), then the length of the solid is equal to 2(x + x), or 2(2x) = 4x. Therefore, (4x)(x)(x) = 108, 4x³ = 108, x³ = 27, and x = 3. If the width and height of the solid are each 3 inches, then the length of the solid is 2(3 + 3) = 2(6) = 12 inches.
22. d.   Use the Pythagorean theorem: 8² + x² = (8)², 64 + x² = 320, x² = 256, x = 16 (x cannot equal –16 because the side of a triangle cannot be negative).
23. b.   The measures of the angles of a triangle add to 180 degrees. Therefore, 3x + 4x + 5x = 180, 12x = 180, and x = 15.
24. e.   To find the distance between two points, use the distance formula:



25. c.   The midpoint of a line segment is equal to the average of the x values of the endpoints and the average of the y values of the endpoints:



26. c.   The slope of a line is the difference between the y values of two points divided by the difference between the x values of those two points: [–5 – (–5)] ÷ –5 – 5 = = 0.