Problem 42 - Algebra A Combined Function

Question

A quadrilateral is a polygon with 4 sides. The sum of the measures of the 4 angles in a quadrilateral is \(360^{\circ} .\) If the measures of the angles of a quadrilateral are consecutive odd integers, find the measures.

Step-by-Step Solution

Answer

The angles of the quadrilateral are \(87^\circ, 89^\circ, 91^\circ,\) and \(93^\circ.\)

Step 1: Define Variables for Angles

Let's define the measures of the four angles as consecutive odd integers. The smallest angle can be denoted as \(x\). The next consecutive odd integers can be expressed as \(x+2\), \(x+4\), and \(x+6\).

Step 2: Write the Equation for the Angles

Using the fact that the sum of the angles in a quadrilateral is \(360^\circ\), we can write the equation: \[ x + (x+2) + (x+4) + (x+6) = 360. \]

Step 3: Simplify the Equation

Combine like terms in the equation: \[ 4x + 12 = 360. \]

Step 4: Solve for x

First, subtract 12 from both sides of the equation: \[ 4x = 348. \]Next, divide both sides by 4: \[ x = 87. \]

Step 5: Calculate the Measures of Each Angle

Now that we have \(x = 87\), calculate each angle using the expressions for consecutive odd integers: - First angle: \(87\)- Second angle: \(87 + 2 = 89\)- Third angle: \(87 + 4 = 91\)- Fourth angle: \(87 + 6 = 93\).

Key Concepts

Consecutive Odd Integers

Consecutive odd integers are numbers that follow each other in sequence, each differing by 2 from the next so they are always odd. For example, if one number is 5, a consecutive odd integer would be 7, and the next would be 9. This pattern of increasing by 2 continues indefinitely.

When dealing with relationships involving consecutive odd integers, it's helpful to define each integer in terms of a single variable.
- Let the smallest integer be represented by a variable like \( x \). - Then, the next odd integers can be expressed in terms of \( x+2 \), \( x+4 \), and so on.

This approach makes it easy to set up equations when solving problems involving consecutive integers, like when you need to find missing values if you're given a set sum of these integers.

Sum of Angles in Quadrilateral

The sum of the angles in a quadrilateral refers to the total measure of the internal angles in any four-sided polygon. No matter the shape of the quadrilateral, the sum of its internal angles is always \( 360^{\circ} \). This is a fundamental property of quadrilaterals.

To help remember this, here's a quick reasoning: Consider dividing the quadrilateral into two triangles. Each triangle has angle measures that sum to \( 180^{\circ} \). Therefore, two triangles together yield a sum of \( 360^{\circ} \).

In practical problems, this property can be utilized to create and solve equations. For example, if you know the angles are consecutive odd integers, you can set these as \( x, x+2, x+4, \) and \( x+6 \). Then, you use the sum \( x + (x+2) + (x+4) + (x+6) = 360 \) to find their actual measure. This setup makes it a straightforward task to determine each angle.

Polygon

A polygon is a two-dimensional geometric figure that is composed of straight line segments. Each connecting pair of segments forms a vertex, and these segments together enclose a space, forming a closed shape.

There are many types of polygons characterized by the number of sides or vertices. For instance:

• Triangle: 3 sides
• Quadrilateral: 4 sides (including squares, rectangles, trapezoids)
• Pentagon: 5 sides

A key concept related to polygons is that as the number of sides increases, the measure of the sum of the interior angles also increases.

For any \( n \)-sided polygon, the sum of its interior angles is given by the formula \( (n-2) \times 180^{\circ} \). This helps in calculating the angle measures for numerous-sided polygons, not limited to quadrilaterals alone.