If necessary, review:

**Points, Lines, and Planes Practice Test**

**Directions:** A good score is eight correct.

1. Which of the following statements is false?

(a) If a line is normal to one of two parallel planes, and that line does not lie in either plane, then that line is normal to the other of the two parallel planes

(b) If a line is parallel to one of two parallel planes, and that line does not lie in either plane, then that line is parallel to the other of the two parallel planes

(c) In 3D space, there cannot exist three different planes, each of which is perpendicular to the other two

(d) In 3D space, there can exist infinitely many different planes, each of which is parallel to all the others

2. Suppose three different planes *X* _{1} , *X* _{2} , and *X* _{3} intersect in a single line *L* . Imagine a fourth plane *X* _{4} that is perpendicular to all three of the planes *X* _{1} , *X* _{2} , and *X* _{3} . Which of the following statements is true?

(a) Line *L* is normal to plane *X* _{4}

(b) Line *L* is parallel to plane *X* _{4}

(c) Any line in plane *X* _{4} is a skew line relative to *L*

(d) There can exist no plane *X* _{4} with the characteristics described

3. Suppose that in a certain 3D universe called *U* , it is impossible to find a plane *Y* that is parallel to a certain plane *X* . From this, we can conclude that

(a) all planes in *U* are flat

(b) half of the planes in *U* are curved

(c) *U* is non-Euclidean

(d) time is warped in *U*

4. Suppose two different planes *X* _{1} and *X* _{2} intersect in a single line *L* . Consider a line *M* that is normal to both planes *X* _{1} and *X* _{2} . Which of the following statements is true?

(a) Line *M* must be normal to either *X* _{1} or *X* _{2}

(b) Line *M* must lie in either *X* _{1} or *X* _{2}

(c) Line *M* must intersect line *L*

(d) There can exist no line *M* with the characteristics described

5. Imagine that you are in a huge warehouse in which the floor and ceiling are both level and flat. You place a flashlight on the floor so the center of its beam shines upward at a 20° angle relative to the floor. The beam strikes the ceiling some distance away. At what angle, with respect to a normal line to the ceiling, does the center of the beam strike the ceiling?

(a) 20°

(b) 70°

(c) 160°

(d) It is impossible to tell without more information

6. Imagine you are in a huge warehouse in which the floor and ceiling are both level and flat, the ceiling is made of glass, and the ceiling forms the roof of the warehouse. You place a flashlight on the floor so the center of its beam shines upward at a 40° angle relative to the floor. The beam strikes and passes through the glass roof some distance away. At what angle, with respect to the plane of the roof, will the center of the beam emerge from the glass?

(a) 40°

(b) 50°

(c) 130°

(d) It is impossible to tell without more information

7. Suppose a closed plane region, having the shape of an octagon, has interior area equal to *k* square units. Suppose the perimeter of the region is *m* units. The interior area of the open region (the octagon not including the outer boundary) is

(a) *k* square units

(b) *k − m* square units

(c) *k − m* units

(d) impossible to determine without more information

8. A plane can be uniquely defined by

(a) three points

(b) two intersecting lines

(c) a line and a point not on that line

(d) any of the above

9. Suppose two different planes *X* _{1} and *X* _{2} intersect in a single line *L* . Consider a line *M* that is parallel to both planes *X* _{1} and *X* _{2} . Which of the following statements is true?

(a) Line *M* must lie in either *X* _{1} or *X* _{2}

(b) Line *M* must lie in both *X* _{1} and *X* _{2}

(c) Line *M* must lie outside both *X* _{1} and *X* _{2}

(d) There can exist no line *M* with the characteristics described

10. Suppose two planes intersect at an angle of 70°. This is the more common of two ways the intersection angle can be expressed. The less common value for the intersection angle of these two planes is

(a) 20°

(b) 180°

(c) 290°

(d) none of the above

**Answers**

1. c

2. a

3. c

4. d

5. b

6. a

7. a

8. d

9. c

10. d

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