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Chapter 7: Problem 72

Find the measure of an angle such that the sum of the measures of its complement and its supplement is \(160^{\circ}\)

Short Answer

Expert verified
The measure of the angle is \( 55^{\circ} \).

Step by step solution

01

Define the variables

Let the measure of the angle be \( x \). Its complement is \( 90^{\circ} - x \) and its supplement is \( 180^{\circ} - x \).
02

Set up the equation

According to the problem, the sum of the measures of the complement and supplement is \( 160^{\circ} \).Therefore, \( (90^{\circ} - x) + (180^{\circ} - x) = 160^{\circ} \).
03

Simplify the equation

Combine like terms: \( 90^{\circ} - x + 180^{\circ} - x = 160^{\circ} \)\( 270^{\circ} - 2x = 160^{\circ} \).
04

Isolate the variable

Subtract 270 from both sides: \( 270^{\circ} - 2x - 270^{\circ} = 160^{\circ} - 270^{\circ} \)\( -2x = -110^{\circ} \).
05

Solve for \( x \)

Divide both sides by -2: \( x = \frac{-110^{\circ}}{-2} \)\( x = 55^{\circ} \).

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Key Concepts

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

Complementary Angles
Complementary angles are a pair of angles whose measures add up to 90 degrees. For example, if one angle measures 30 degrees, the other must measure 60 degrees to be complementary.
Complementary angles are found often in problems related to geometry and right triangles. When angles are complementary, they often help in solving for unknown angle measures in these shapes. Remember, it's key to identify which angles are complementary, as this relationship simplifies solving for unknown variables.
Supplementary Angles
Supplementary angles are a pair of angles whose measures add up to 180 degrees. For instance, if one angle measures 110 degrees, the other must be 70 degrees to be supplementary.
Supplementary angles frequently appear in problems involving linear pairs and straight lines. Recognizing when angles are supplementary is crucial for geometric proofs and solving for unknown angles in various figures. Similar to complementary angles, knowing this relationship simplifies the problem-solving process.
Solving Linear Equations
Solving linear equations involves finding the value of the variable that makes the equation true. In these equations, you will typically see a straightforward relationship between variables and constants, like in the form of ax + b = c.
Here's the general process:
  • Combine like terms on each side of the equation.
  • Use the addition or subtraction property of equality to isolate the variable term on one side of the equation.
  • Finally, use the multiplication or division property of equality to solve for the variable.
Practicing these steps helps in handling more complicated algebraic expressions.
Variables in Algebra
Variables are symbols used to represent unknown values in algebraic expressions and equations. They can be letters like x, y, or z.
Understanding variables is fundamental in algebra, as they allow us to formulate and solve an array of problems. When working with variables, it's important to:
  • Define what the variable represents clearly.
  • Set up equations based on the given problem conditions.
  • Solve for the variables using appropriate algebraic techniques.
With practice, manipulating and solving for variables becomes second nature.

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Most popular questions from this chapter

Without graphing, do the following for each system of equations. (a) Describe each system. (b) State the number of solutions. (c) Is the system inconsistent, are the equations dependent, or neither? \(2 x-y=6\) \(-10 x+5 y=30\)

A circle has an equation of the following form. $$x^{2}+y^{2}+a x+b y+c=0$$ It is a fact from geometry that given three noncollinear points-that is, points that do not all lie on the same straight line-there will be a circle that contains them. For example, the points \((4,2),(-5,-2),\) and (0,3) lie on the circle whose equation is shown in the figure. To find \(a n\) equation of the circle passing through the points $$(2,1),(-1,0), \text { and }(3,3)$$ Let \(x=2\) and \(y=1\) in the general equation \(x^{2}+y^{2}+a x+b y+c=0\) to find an equation in \(a, b,\) and \(c\)

Solve each system by the elimination method. Check each solution. $$ \begin{array}{l} 2 x+3 y=0 \\ 4 x+12=9 y \end{array} $$

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Without graphing, do the following for each system of equations. (a) Describe each system. (b) State the number of solutions. (c) Is the system inconsistent, are the equations dependent, or neither? \(2 x-y=4\) \(y+4=2 x\)
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