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Chapter 2: Problem 24

Is a polyhedron necessarily a prism, if two of its faces are congruent polygons with respectively parallel sides, and all other faces are parallelograms? (First allow non-convex polyhedra.)

Short Answer

Expert verified
Yes, the polyhedron is a prism.

Step by step solution

01

Understanding the Problem

A polyhedron with two congruent polygonal faces with parallel sides and all other faces as parallelograms is given. We need to determine if this implies the polyhedron is a prism.
02

Defining a Prism

A prism is a polyhedron with two parallel faces (bases) that are congruent, and all other faces (lateral faces) are parallelograms. These lateral faces are parallelograms connecting each side of the base.
03

Analyzing the Condition

The problem states that there are two congruent polygonal faces with parallel sides and that all other faces are parallelograms. It's important to check if this satisfies the definition of a prism.
04

Verifying Parallelism of Bases

In a prism, the two congruent bases must be parallel. The condition provided only speaks of congruent polygonal faces with parallel sides, suggesting these could serve as the bases of a prism.
05

Testing All Other Faces for Compatibility

The condition states that all other faces are parallelograms. In a prism, these lateral parallelograms arise from the edges of the base and connect corresponding vertices of the bases.
06

Confirming Identity as a Prism

Since the polyhedron has two congruent polygonal faces with respectively parallel sides and all other faces as parallelograms, it matches the structural description of a prism.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polyhedra: The Building Blocks
In the world of geometry, a polyhedron is a solid figure with flat faces. These faces are polygons, and they come together to form a three-dimensional shape.
A polyhedron is made up of:
  • Vertices: These are the corner points where the edges meet.
  • Edges: The line segments where two faces meet.
  • Faces: The flat surfaces of the polyhedron, which are polygons.
Polyhedra can take on many different forms, including prisms and pyramids, each with unique properties.
Prisms are a specific type of polyhedron, characterized by having two congruent and parallel polygonal bases. All other faces of a prism are parallelograms. Understanding this helps clarify why a polyhedron with congruent polygonal faces and parallelogram sides fits the definition of a prism.
Congruent Polygons: Twins in Shape and Size
Congruent polygons are polygons that are exactly the same in shape and size. This means their corresponding sides and angles are equal.
When discussing polyhedra and prisms, congruent polygons are important because they often make up the bases of these shapes. For a prism, the bases must be congruent because this ensures the structure is uniform and follows the necessary geometric rules.
Key characteristics of congruent polygons include:
  • They have an equal number of sides and angles.
  • All corresponding sides are equal in length.
  • All corresponding angles are equal in measure.
The significance of having congruent polygonal bases in a prism lies in ensuring that the shape can be properly enclosed by the parallelograms that form the sides. This is crucial in maintaining the geometric integrity of the prism.
Parallelograms: The Essential Side Faces
Parallelograms are four-sided figures with opposite sides that are parallel and equal in length. Their properties make them the quintessential shape for the sides of prisms.
Here are some important properties of parallelograms:
  • The opposite sides are equal in length and parallel.
  • The opposite angles are equal.
  • Adjacent angles are supplementary, meaning they add up to 180 degrees.
In the context of prisms, parallelograms serve a vital role. They are the lateral faces connecting the two bases.
The nature of their parallel sides harmonizes perfectly with the parallel edges of the prism's bases, ensuring the structure's consistency and stability.
This is why in the original exercise, it was significant that all non-basal faces were parallelograms, confirming the identity of the polyhedron as a prism.

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Most popular questions from this chapter

Classification of regular polyhedra. Let us take into account that a convex polyhedral angle has at least three plane angles, and that their sum has to be smaller than \(4 d(\S 48)\). Since in a regular triangle, every angle is \(\frac{2}{3} d\), repeating it 3,4 , or 5 times, we obtain the angle sum smaller than \(4 d\), but repeating it 6 or more times, we get the angle sum equal to or greater than \(4 d\). Therefore convex polyhedral angles whose faces are angles of regular triangles can be of only three types: trihedral, tetrahedral, or pentahedral. Angles of squares and regular pentagons are respectively \(d\) and \(\frac{6}{5} d\). Repeating these angles three times, we get the sums smaller than \(4 d\), but repeating them four or more times, we get the sums equal to or greater than \(4 d\). Therefore from angles of squares or regular pentagons, only trihedral convex angles can be formed. The angles of regular hexagons are \(\frac{4}{3} d\), and of regular polygons with more than 6 sides even greater. The sum of three or more of such angles will be equal to or greater than \(4 d\). Therefore no convex polyhedral angles can be formed from such angles. It follows that only the following five types of regular polyhedra can occur: those whose faces are regular triangles, meeting by three, four or five triangles at each vertex, or those whose faces are either squares, or regular pentagons, meeting by three faces at each vertex.

Prove that if all lateral faces of a pyramid form congruent angles with the base, then the base can be circumscribed about a circle.

A pyramid with the altitude \(h\) is divided by two planes parallel to the base into three parts whose volumes have the ratio \(l: m: n\). Find the distances of these planes from the vertex.

Determine the number of planes of symmetry of a regular pyramid with \(n\) lateral faces. 131\. Let three figures \(\Phi, \Phi^{\prime}\), and \(\Phi^{\prime \prime}\) be symmetric: \(\Phi\) and \(\Phi^{\prime}\) about a plane \(P\), and \(\Phi^{\prime}\) and \(\Phi^{\prime \prime}\) about a plane \(Q\) perpendicular to \(P\). Prove that \(\Phi\) and \(\Phi^{\prime \prime}\) are symmetric about the intersection line of \(P\) and \(Q\). 132\. What can be said about the figures \(\Phi\) and \(\Phi^{\prime \prime}\) of the previous problem if the planes \(P\) and \(Q\) make the angle: (a) \(60^{\circ} ?\) (b) \(45^{\circ} ?\)

Prove that a polyhedron is convex if and only if every segment with the endpoints in the interior of the polyhedron lies entirely in the interior.
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