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Chapter 5: Problem 48

Find the angle that is supplementary to it. $$112^{\circ}$$

Short Answer

Expert verified
The supplementary angle to 112 degrees is \(180^{\circ} - 112^{\circ} = 68^{\circ}\)

Step by step solution

01

Understanding what Supplementary Angles Mean

You need to recognize that supplementary angles are two angles that add up to 180 degrees. This is a key fact used for this problem.
02

Identifying the Given Angle

From the question, we know that one of the angles is 112 degrees. We are asked to find the angle that is supplementary to this one.
03

Calculating the Supplementary Angle

To find the supplementary angle, subtract the given angle from 180 degrees. In this case, we subtract 112 degrees from 180 degrees. This could be written mathematically as \(180^{\circ} - 112^{\circ}\).

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Key Concepts

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

Angle Calculation
Calculating angles can be a fun puzzle! One of the most common calculations in geometry involves finding missing angles like in our task. To solve these problems, you should be familiar with mathematical operations like subtraction and know the properties of angles.

In the case of supplementary angles, the key operation is subtraction. You'll subtract the known angle from 180掳, our magic number for supplementary angles. This calculation is straightforward:
  • Identify the given angle (here, 112掳).
  • Subtract it from 180掳 to find the missing angle.
In our example, subtracting gives the other angle: \(180^{\circ} - 112^{\circ} = 68^{\circ}\). So, the supplementary angle is 68掳. Angle calculation is not just useful for homework; it helps in understanding geometry as a whole!
Complementary and Supplementary Angles
In geometry, angles can pair up in interesting ways. Complementary angles, for example, are two angles that add up to 90 degrees. But this exercise focuses on supplementary angles which sum up to 180 degrees.

Understanding this difference is essential:
  • Complementary angles form a right angle together (90掳).
  • Supplementary angles form a straight line together (180掳).
Both types of angle pairs are based on their sum. This characteristic is a fundamental part of geometry and is extensively used in solving problems. Recognizing the type of angle relation is the first step in determining how to find the missing angle. So, if you're asked to find a supplementary angle, you know the total has to be 180掳, guiding your calculations.
Basic Geometry Concepts
Geometry is like a language of space, and understanding its basic concepts makes solving problems much easier. Fundamental notions in geometry include points, lines, and angles.

An angle is essentially a measure of rotation between two intersecting lines or rays. The unit of measurement for angles is degrees.
  • Right angles equal 90掳.
  • Straight angles are 180掳.
  • Complete angles circle back to 360掳.
Knowing these concepts helps in categorizing angles and understanding their relationships, such as complementary and supplementary angles. Mastery of these basics is crucial for solving more complex mathematical problems and understanding the spatial relations and properties in the world around us. Therefore, always reinforce these core ideas to perform well in geometry.

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Most popular questions from this chapter

Consider an angle \(\theta\) in standard position whose vertex coincides with the center of a circle of radius \(r .\) The portion of the circle bounded by the initial side and the terminal side of the angle \(\theta\) is called a sector of the circle. (a) If \(A\) is the area of the circle, then \(A_{s}=A \frac{\theta}{2 \pi}\) represents the area of the sector because \(\frac{\theta}{2 \pi}\) gives the fraction of the area covered by the sector. Show that the area of a sector, \(A_{s},\) is \(A_{s}=\frac{r^{2} \theta}{2} .\) Here theta is in radians. (b) Find \(A_{s}\) if \(\theta=\frac{\pi}{3}\) and \(r=12\) inches.

Derive the Pythagorean identity \(1+\cot ^{2} t=\csc ^{2} t\)

Graph the given pair of functions in the same window. Graph at least two cycles of each function, and describe the similarities and differences between the graphs. $$f(x)=\sec (2 x) ; f(x)=2 \sec (x)$$

Find the radian measure of an angle in standard position that is generated by the specified rotation. Quarter of a full revolution clockwise

Graph at least two cycles of the given functions. $$f(x)=-\tan \left(x-\frac{\pi}{3}\right)-1$$
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