Compare the polygon CDEFG and C'D'E'F'G'.

Sug. Show how C D and C' D', D E and D' E', etc., are related, and also how / C D E and C'D' E', DE F and Z D' E' F', etc., are related.

Can you now show the relation between the polygons?

472.

Cor. (1) Prove what the section is when the prism is cut by a plane || to the base.

(2) How do right sections compare?

Suppose A D' represent any oblique prism and F G H I J a right section.

Can you find an expression for its lateral surface?

Sug. What are the faces? How do the edges compare? How does the area of any face compare with a rectangle hav

ing the same base and an equal altitude? May we consider a lateral edge as base of the parallelograms?

If, for example, we take A A' as the base of the parallelogram A' B, what is its altitude?

How does the sum of the altitudes of the faces compare with the perimeter of a right section.

Complete proof and write the generalization. Call the statement Prop. II.

474.

Cor. How find the area of a right prism?

475.

PROPOSITION III.

JE

Given: In the prisms A d, A' d' the faces B E, Bc, Ba respectively equal to B' E', B'c', B'a', and similarly arranged Compare the volumes of the prisms.

Sug. Compare any two corresponding triedrals, as B and B'. Can you make A D coincide with A' D'? A b coincide with A'b? Bc coincide with B' d'? a b with a'b'? bc with b' c? Show that a d coincides with a' d'.

Finish the proof and write the generalization. Call this Prop. III.

476.

Cor. 1. When are two right prisms equal?

477.

Cor. II. Suppose the figures above are truncated prisms and the conditions the same. Compare the solids.

478.

PROPOSITION IV.

Given. The oblique prism A D' and F G H I J a right section.

B

Can you find a right prism, having for its base a right section of the oblique prism and an altitude

equal to a lateral edge of the oblique prism, which shall be equivalent to the given oblique prism?

Sug. Extend the lateral edges A A', B B', etc., making F FA A'. At F' pass a plane 1 F F'. How is this plane related to FI? What is FI? A' I'? A I? Compare B B'

with G G'.

How do B G and B'G' campare? A B and A' B? FG and F'G'? What is A G? A' G? Compare them.

In a similar manner compare B H and B' H'.

Compare A B C with A'B'C' and base B E with base B' E'. Compare the triedral angle A B GC with the triedral A' B' G' C'.

Compare the truncated prisms A I and A' I'. Can you now complete the proof to the answer of the original question? Write the proposition, and call it Prop. V.

A parallelopiped is a prism whose bases are parallelograms.

480.

A right parallelopiped is one whose lateral edges are perpendicular to the bases.

481.

A rectangular parallelopiped is a right parallelopiped having all its faces rectangles. How would you define a cube? (See Fig. S485.)

EXERCISE.

438. Draw a right parallelopiped whose bases are, (1) trapeziums, (2) trapezoids, (3) rhomboids, (4) rhombuses.

Given any parallelopiped, A G, and consider A H and B G the bases. Compare the opposite faces.

[Hint.-Compare E F, H G, D C, A B; also B C, G F, E H, A D. Compare Z FEH and BA D, ZE H G and ZA DC. Compare faces AC and E G.]

What can you say of the opposite faces of a parallelopiped?

Write the general statement, and call it Prop. V.

EXERCISE.

439. Let P Q be a section formed by passing a plane through the parallelopiped A G, cutting only 2 pairs of opposite sides.

Prove what the section is.