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Chapter 7: Problem 4

Without looking them up, write the definitions of the following geometric terms. Rectangle.

Short Answer

Expert verified
A rectangle is a quadrilateral with equal opposite sides and all angles as 90 degrees.

Step by step solution

01

Basic Shape Identification

A rectangle is a four-sided polygon, which is a special type of quadrilateral. To begin, identify that a rectangle is a shape with four sides.
02

Opposite Sides Analysis

In a rectangle, all opposite sides are equal in length. This means the rectangle has two pairs of equal-length sides (e.g., the length of one pair is equal, and the length of the other pair is equal).
03

Internal Angles

A defining characteristic of a rectangle is that all its internal angles are right angles, each measuring exactly 90 degrees. This means a rectangle is both an equilateral (in terms of angles) and a right-angle equilateral.
04

Conclusion of Definition

Combining all the facts, a rectangle is a quadrilateral with opposite sides equal and all angles measuring 90 degrees.

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Key Concepts

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

Quadrilateral
A quadrilateral is a specific type of polygon that always has four edges and four vertices. In geometric terms, this simply means it's a shape with four straight sides. Each angle in a quadrilateral adds up to a total of 360 degrees. This four-sided polygon is a broad category, which includes not only rectangles but also squares, rhombuses, and trapezoids.
  • Quadrilaterals can vary greatly in their angles and side lengths.
  • Each of them is classified by specific properties, such as equal sides or specific angles.
Understanding quadrilaterals is fundamental to recognizing and distinguishing shapes like rectangles and squares, which are both specialized types within this category.
Opposite Sides
In the context of rectangles, the term 'opposite sides' refers to the pairs of sides that do not meet at a vertex. These pairs of sides lie directly across from each other on the rectangle.
  • A rectangle has two pairs of opposite sides, and in each pair, the sides are equal in length.
  • This feature is crucial, and it helps to clearly differentiate rectangles from other similar shapes, such as parallelograms or rhombuses.
Seeing how opposite sides function can make understanding rectangle properties much clearer. A rectangle's unique symmetry is largely due to this characteristic of having equal-length opposite sides.
Right Angles
Right angles are a defining property of rectangles. Each internal angle in a rectangle measures exactly 90 degrees, which is the definition of a right angle. This ensures that the sides of a rectangle meet perpendicularly.
  • Having four right angles is part of what makes a rectangle a specific type of quadrilateral.
  • This characteristic helps the rectangle maintain its consistent shape and is what gives rectangle their structured form.
Learning about right angles and their significance is key to understanding the rigorous symmetry and definition of rectangles. This aspect also highlights why rectangles are widely used in design, architecture, and everyday items due to their reliable shape.

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Most popular questions from this chapter

Quadrilateral PRIZ is an isosceles trapezoid with diagonals \(\overline{\mathrm{PI}}\) and \(\overline{\mathrm{ZR}}\). Use this information to tell whether each of the following statements must be true, may be true, or appears to be false. CAN'T COPY THE GRAPH $$\triangle \mathrm{PZI} \cong \triangle \mathrm{RIZ}$$

Tell whether each of the following statements is true or false. If a quadrilateral is a parallelogram, then it is equilateral.

Given: In quadrilateral SNAP, $$\overline{\mathbf{S N}} \| \overline{\mathbf{A P}}$$ \(\sphericalangle \mathrm{N}\) and \(\sphericalangle \mathrm{P}\) are supplementary. $$\text { Prove: } \angle \mathrm{S}=\angle \mathrm{N}$$ (IMAGE CAN'T COPY)

Quadrilateral HUGO is a square. Why is HUGO a parallelogram?

Tell whether each of the following statements is true or false. If a quadrilateral is not a parallelogram, then its diagonals do not bisect each other.
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