Question

Find the measure of one interior angle of a regular quadrilateral.

Short answer

Each interior angle of a regular quadrilateral measures 90°.

Step-by-step solution

Understand the Problem

A regular quadrilateral is a four-sided polygon where all sides and angles are equal. The problem asks for the measure of one interior angle of this shape.

Recall the Formula for Interior Angles Sum

The sum of interior angles of any polygon with \( n \) sides is given by \((n-2) \times 180^\circ\). For a quadrilateral, \( n = 4 \), so the sum is \((4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ\).

Calculate One Interior Angle

Since the quadrilateral is regular, all four interior angles are equal. To find the measure of one interior angle, divide the total sum of the angles by the number of angles: \( \frac{360^\circ}{4} = 90^\circ \).

Key concepts

Regular Quadrilateral

A regular quadrilateral is a specific type of polygon that has four equal sides and four equal angles. Imagine a perfect square – this is a common example of a regular quadrilateral. All the angles in a regular quadrilateral are right angles, each measuring exactly 90 degrees. Regular quadrilaterals hold some unique properties:

• They are always symmetrical, meaning they can be divided into two equal halves through lines known as axes of symmetry.
• Each side is of the same length, creating an equilibrium in structure and appearance.

Understanding these properties helps in identifying a regular quadrilateral and calculating its angles easily.

Polygon Angles Formula

The Polygon Angles Formula is essential for determining the sum of interior angles of any polygon. This formula is given by \( (n-2) \times 180^\circ \), where \( n \) represents the number of sides. This formula comes from the understanding that any polygon can be divided into triangles, and since each triangle has a sum of interior angles equaling 180 degrees, multiplying the number of triangles by 180 degrees gives the sum of all angles.

For example, in a quadrilateral where there are four sides (\ \( n = 4 \ \)), the formula becomes \( (4-2) \times 180^\circ = 360^\circ \). This result means no matter what type of quadrilateral you're dealing with, the sum of its interior angles will always be 360 degrees. This consistency makes this formula a reliable tool for solving various geometry problems.

Quadrilaterals

Quadrilaterals are four-sided polygons that come in various shapes and sizes. Familiar examples include squares, rectangles, trapezoids, and rhombuses. Despite their differing appearances, they share some common characteristics:

• All quadrilaterals have four sides and four vertices.
• The sum of their interior angles is always 360 degrees, as derived from the Polygon Angles Formula.
• They can be categorized into regular and irregular based on their side lengths and angle measurements.

Understanding the different types of quadrilaterals helps in identifying their properties and solving geometric problems efficiently. For instance, knowing that all sides and angles are equal in a regular quadrilateral like a square, helps calculate the angles easily. On the other hand, irregular quadrilaterals might require different approaches based on their unique characteristics.