John is shown a picture of a cube. He is asked to tell how many vertices, faces, and edges it has. His answer follows:
Faces: $3$
Vertices: $7$
Edges: $9$
(a) Explain what John knows about geometric definitions and is identifying correctly.
(b) Identify what John is doing incorrectly.
(c) How would you guide him in approaching this problem in the future?

A soccer ball, like the one below, is a type of polyhedron sometimes called a buckyball. It has 32 faces, of which 20 are hexagons and 12 are pentagons. If a buckyball has 60 vertices, how many edges does it have?

Tianna is given a set of geometric solids. She correctly identifies the various pyramids but calls all of the prisms in the set rectangular prisms. The set Tianna was given actually contains a triangular prism, a rectangular prism, a pentagonal prism, and a hexagonal prism.
(a) What is Tianna thinking when she improperly names the prisms?
(b) How would you guide her so that in the future she is able to properly identify geometric solids?

a. Determine the measure of an angle whose measure is twice that of its complement.
b. If two angles of a triangle are complementary, what is the measure of the third angle?

Josh is going to draw a net of a triangular prism. How many rectangles should there be in his drawing?

Classify each of the following statements as true or false. If false, tell why.
a. The bases of all cones are circles.
b. A cylinder has only one base.
c. All lateral faces of an oblique prism are rectangular regions.
d. All regular polyhedra are convex.
e. A cube has congruent edges.
f. Any pair of opposite faces of a rectangular prism can be bases.

The base of a prism has n sides.
a. How many lateral faces does it have?
b. How many faces does it have?

Draw two isosceles triangles that are not congruent but that have a perimeter of 15 cm each.

Suppose $\triangle A B C \sim \triangle B C A$. What more can you say about $\triangle A B C ?$

Three liquids of different densities are mixed. Because they are not miscible (do not form a homogeneous solution with one another), they form discrete layers, one on top of the other. Sketch the result of mixing carbon tetrachloride $\left(\mathrm{CCl}_4, d=1.58 \mathrm{~g} / \mathrm{cm}^3\right)$, mercury $\left(d=13.546 \mathrm{~g} / \mathrm{cm}^3\right)$, and water $\left(d=1.00 \mathrm{~g} / \mathrm{cm}^3\right)$.

If three lines all meet in a single point, how many pairs of vertical angles are formed?

Millie claims that a rhombus is regular because all of its sides are congruent. How do you respond?

Draw freehand pictures of the figures that follow, using dashed lines to indicate hidden edges. Don’t just copy— transfer the image in your mind to paper.
(a) Pentagonal right prism
(b) Oblique hexagonal prism
(c) Octahedron
(d) Right circular cone

Explain how, through paperfolding, you would show each of the following statements.
a. In an isosceles triangle, the angles opposite congruent sides are congruent.
b. If two angles of a triangle are congruent, the triangle is isosceles.
c. In an equilateral triangle, all the interior angles are congruent.
d. An isosceles trapezoid has two pairs of congruent angles.