1. The following figure is a box in which the top and bottom are rectangles and BF perpendicular to plane FGH. Answer the following: a. Find the intersection ofBH and plane DCG. b. Name two pairs of perpendicular planes. c. Name two lines that are perpendicular to plane EFH. d. What is the measure of dihedral angle D-HG-F?

a) Point H

b) BCD  DCG, and EFG  BFG

c) DH and BF

d) 90 degrees

5. In each of the following, relationships among marked angles are given below the figure. Find the measures of the marked angles. C D O A B m(DOC) _ m(BOA) 34 b. O A C B m(AOB) is 30_ less than 2 m(BOC) c. C O A B y x m(AOB) – m(BOC) = 50_

a) DOC = 15o BOA = 45o

b) AOB = 50o BOC = 40o

c) AOB = 115o BOC = 65o

7. Within the classroom, identify a physical object with the following: a. Parallel lines b. Parallel planes c. Skew lines d. Right angles

a) floor pattern

b) ceiling and floor

c) diagonal line on floor and opposite diagonal line on ceiling

d) floor and wall

10. A student claims that if any two planes that do not intersect are parallel, then any two lines that do not intersect should also be parallel. How do you respond?

No, it’s possible for two lines to be skew in three dimensions.

In three dimensions, planes (2D objects) that do not intersect must be parallel. Similarly, in two dimensions, lines (1D objects) that do not intersect are parallel. But in three dimensions, lines can be skew, and planes can be “skew” in 4 dimensions.

1. Determine for each of the following which of the figures (if any) labeled (1) through (10) can be classified under the given terms: a. Isosceles triangle b. Isosceles but not equilateral triangle c. Equilateral but not isosceles triangle d. Parallelogram but not a trapezoid e. A trapezoid but not a parallelogram f. A rectangle but not a square g. A square but not a rectangle h. A square but not a trapezoid i. A rhombus but not a kite j. A rhombus k. A kite

a)1, 2

b)1

c)2

d)none

e)5

f)7

g)none

h)6

i)none

j)8

9. One student says, “My sister’s high school geometry book talked about equal angles. Why don’t we use the term ‘equal angles’ instead of ‘congruent angles’?” How do you reply?

Strictly speaking, equal angles means they are literally the same angle, formed by the same two rays. Congruent means their measure is equal, even though they are not the identically same angle.

2. In the following figures, find the measures of the angles marked x and y: a.70° B A C D x y AB || CD b. x 125? 42?

a) x = 50, y = 60

b) x = 83

5. Find the sum of the measures of the marked angles in each of the following figures: 2 2 1 3 1 3 6 4 6 4 a. 5 b. 5

a) 360

b) 360

1. a. If one angle of a triangle is obtuse, can another also be obtuse? Why or why not?

No, because the sum of the angles must equal 180 degrees.

b. If one angle in a triangle is acute, can the other two angles also be acute? Why or why not?

Yes, for example all three can be 60 degrees.

c. Can a triangle have two right angles? Why or why not?

No, because the remaining angle would have measure zero.

d. If a triangle has one acute angle, is the triangle necessarily acute? Why or why not?

No, one of the other angles may be obtuse, making it an obtuse triangle.

4. In the following figure, the legs of the ladder are congruent. If the ladder makes an angle of120 degrees with the ground, what is x? Explain your reasoning.

x = 120 – 90 = 30 degrees.

2. The following are pictures of solid cubes lying on a flat surface. In each case, determine the number of cubes in the stack and the number of faces that are glued together. XX X XXX XXX XXX A. B.

a) 7 cubes, 8 glued sides

b) 28 cubes, 53 glued sides

8. Given a tetrahedron, explain how to obtain each of the following shapes by slicing the tetrahedron by a plane. Describe a procedure for getting each shape. a. An equilateral triangle b. A scalene triangle c. A rectangle

a) slice through one corner of the tetrahedron so the cut is equidistant from the vertex on all faces.

b) slice through a corner at an oblique angle

c) can’t be constructed.

2. A circle can be approximated by a “many-sided” polygon. Use this notion to describe the relationship between each of the following: a. A pyramid and a cone b. A prism and a cylinder

a) a pyramid has a square base. Increasing the number of sides of the base will make the volume approach that of a cone

b) A prism has a square cross section. Imagine increasing this to a pentagon, etc. adding sides. The shape would approach that of a cylinder as the number of sides grows to infinity.