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Chapter 8: Problem 50

Can two supplementary angles both be acute? Why or why not?

Short Answer

Expert verified
No, because their measures must add up to 180 degrees, and two acute angles would sum to less than 180 degrees.

Step by step solution

01

Understand Supplementary Angles

Supplementary angles are two angles whose measures add up to 180 degrees.
02

Understand Acute Angles

An acute angle is any angle that measures less than 90 degrees.
03

Analyze the Conditions

If both angles were acute (each being less than 90 degrees), then their sum would be less than 180 degrees. This does not satisfy the condition of supplementary angles.
04

Conclusion

Therefore, two supplementary angles cannot both be acute because their measures must add up to 180 degrees, which is not possible if both are less than 90 degrees.

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Key Concepts

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

acute angles
An acute angle is an angle that measures less than 90 degrees. This type of angle is sharp and small, like the corners of a slice of pizza.

Here are some key points about acute angles:
  • Any angle measuring from just above 0掳 up to 89.9掳 is considered acute.
  • You will often see acute angles in various geometric shapes like triangles.
  • A triangle with all three acute angles is called an acute triangle.
Why is it important to know this? When solving problems in geometry, understanding that acute angles are smaller than 90 degrees helps you categorize and solve for unknown angle measures more easily.
angle measures
When we talk about angle measures, we are referring to the size of the angle, measured in degrees.

Angles can vary widely in size, and there are special names for ranges of measures:
  • Acute angles are less than 90掳.
  • Right angles are exactly 90掳.
  • Obtuse angles are between 90掳 and 180掳.
  • Straight angles are exactly 180掳.
To solve many geometry problems, you often need to know how to measure and classify different types of angles.

One important concept is that the sum of angles in a triangle is always 180掳. This foundational idea helps in many areas of geometry, especially when working with supplementary and complementary angles.
geometry
Geometry is a branch of mathematics that deals with shapes and the relationships between them. It involves studying points, lines, surfaces, solids, and angles.

Here are some fundamental ideas in geometry:
  • Angles: As discussed, knowing how angles work is crucial in geometry.
  • Lines and Segments: Understanding the properties of lines, line segments, and rays helps in defining shapes and angles.
  • Shapes: Learning about different geometric shapes (like triangles, circles, squares) and their properties is essential.
One of the first things you'll encounter in geometry is learning how to calculate different types of angles, such as supplementary angles. Remember, supplementary angles always add up to 180 degrees, which helps you solve many geometric problems.

For example, since two acute angles both measure less than 90 degrees, they can't be supplementary. This principle is valuable for solving various geometry exercises and understanding the nature of different shapes.

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Most popular questions from this chapter

In the Expanding Your Skills of Section \(8.1,\) we converted U.S. Customary units of area. We use the same procedure to convert metric units of area. This procedure involves multiplying by two unit ratios of length. Example: Converting area Convert \(1000 \mathrm{mm}^{2}\) to square centimeters. $$\text { Solution: } \frac{1000 \mathrm{mm}^{2}}{1} \cdot \frac{1 \mathrm{cm}}{10 \mathrm{mm}} \cdot \frac{1 \mathrm{cm}}{10 \mathrm{mm}}=\frac{1000 \mathrm{mm}^{2}}{1} \cdot \frac{1 \mathrm{cm}^{2}}{100 \mathrm{mm}^{2}}=\frac{1000 \mathrm{cm}^{2}}{100}=10 \mathrm{cm}^{2}$$ convert the units of area, using two factors of the given unit ratio. $$30,000 \mathrm{mm}^{2}= \quad \mathrm{cm}^{2}$$ $$\left(\text { Use } \frac{1 \mathrm{cm}}{10 \mathrm{mm}}\right)$$

Sketch two lines that are perpendicular.

Can two adjacent angles formed by two intersecting lines be complementary, supplementary, or neither?

The measure of an angle is given. Find the measure of the supplement. $$179^{\circ}$$

Byron lays sod in his backyard. Each piece of sod weighs 6 lb 4 oz. If he puts down 50 pieces, find the total weight.
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