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Chapter 5: Problem 32

If an angle measures \(x^{\circ},\) how can we represent its supplement?

Short Answer

Expert verified
The supplement is \[ 180 - x \]

Step by step solution

01

Understanding Supplementary Angles

Supplementary angles are two angles whose measures add up to 180 degrees. Therefore, if you have one angle, its supplement is what you need to add to it to reach 180 degrees.
02

Set Up the Equation

Given an angle measuring x degrees, we need to find another angle that, when added to x, equals 180 degrees. Mathematically, let y be the supplementary angle. Thus, the equation is: \[ x + y = 180 \]
03

Solve for the Supplementary Angle

To find the expression for the supplementary angle (y), solve the equation for y:\[ y = 180 - x \]
04

State the Final Answer

So, the supplement of an angle measuring x degrees is \[ 180 - x \]

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Key Concepts

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

Angle Measurement
Angle measurement is an essential part of geometry. Angles are measured in degrees, which can range from 0 to 360.
Each angle has a specific measure, which helps in understanding the shape, size, and orientation of geometric figures.
There are different types of angles based on their measures:
  • Acute angles: less than 90 degrees
  • Right angles: exactly 90 degrees
  • Obtuse angles: more than 90 but less than 180 degrees
  • Straight angles: exactly 180 degrees

Knowing the measure of an angle is crucial when working with supplementary angles and solving related problems.
Linear Equations
Linear equations are fundamental in algebra and geometry. A linear equation is an equation between two variables that gives a straight line when plotted on a graph.
In the context of supplementary angles, we use a simple linear equation to find the unknown angle.
When you have one angle measuring \(x^{\circ}\), its supplementary angle y can be found using the equation: \[ x + y = 180 \]
This equation states that the sum of the measures of two supplementary angles is always 180 degrees. By rearranging the equation to: \[ y = 180 - x \]
you can easily find the supplementary angle, making it a practical application of linear equations.
Basic Geometry
Basic geometry introduces many fundamental concepts, including angles and their properties.
Understanding basic geometry is necessary for solving problems about shapes, sizes, and the relationships between different geometric figures.
Supplementary angles are a key part of this. They help in understanding how different angles interact within geometric shapes, like triangles and quadrilaterals.
When you learn that \(x + y = 180^{\circ}\), it opens up more complex geometrical transformations and proofs in higher mathematics.
Mastery of basic concepts like this ensures a strong foundation for more advanced studies.

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Most popular questions from this chapter

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\sin \theta, \text { given that } \csc \theta=\frac{\sqrt{24}}{3}$$

Find the five remaining trigonometric finction values for each angle. \(\sec \theta=-4,\) and \(\sin \theta>0\).

Use identities to solve each of the following. Find \(\cos \theta,\) given that \(\sin \theta=\frac{3}{5}\) and \(\theta\) is in quadrant II.

Use the trigonometric finction values of quadrantal angles given in this section to evaluate each expression. An expression such as cot' \(90^{\circ}\) means (cot \(90^{\circ}\) ) \(^{2}\), which is equal to \(0^{2}=0\). $$\cos ^{2}\left(-180^{\circ}\right)+\sin ^{2}\left(-180^{\circ}\right)$$

Work each problem. In these exercises, assume the course of a plane or ship is on the indicated bearing. The bearing from Winston-Salem, North Carolina, to Danville, Virginia, is \(N 42^{\circ}\) E. The bearing from Danville to Goldsboro, North Carolina, is \(S 48^{\circ}\) E. A car driven by Ellen Winchell, traveling at 65 mph, takes \(1.1 \mathrm{hr}\) to go from Winston-Salem to Danville and \(1.8 \mathrm{hr}\) to go from Danville to Goldsboro. Find the distance from Winston-Salem to Goldsboro.
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