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Chapter 1: Problem 7

Explain how the sum of two acute angles could be a Acute b Obtuse c Right

Short Answer

Expert verified
a) Two acute angles can sum to an acute angle if each is less than 45 degrees. b) Two acute angles can sum to an obtuse angle if one or both are greater than 45 degrees such that their sum is greater than 90 degrees. c) Two acute angles can sum to a right angle if each is exactly 45 degrees.

Step by step solution

01

Sum of Two Acute Angles Resulting in an Acute Angle

Keep in mind that an acute angle is less than 90 degrees. So, if two acute angles are each less than 45 degrees, their sum will also be less than 90 degrees and hence the resulting angle is acute.
02

Sum of Two Acute Angles Resulting in an Obtuse Angle

One must remember that an obtuse angle is greater than 90 degrees but less than 180 degrees. As such, if the sum of two acute angles is greater than 90 degrees, then the resulting angle is obtuse. This situation can only occur if at least one of the acute angles is greater than 45 degrees.
03

Sum of Two Acute Angles Resulting in a Right Angle

A right angle is exactly 90 degrees. Thus, if the sum of two acute angles is exactly 90 degrees, then the resulting angle is right. This situation can occur when both acute angles are exactly 45 degrees.

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Key Concepts

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

Angle Sum
Understanding the concept of "angle sum" is crucial when dealing with two or more angles. When we talk about the "sum of angles," we are simply adding the measures of two angles together. This sum can change the type of angle we are dealing with.

Let's explore how this works with two acute angles: an acute angle is any angle less than 90 degrees. When you add the measures of two acute angles:
  • If each angle is less than 45 degrees, their sum remains less than 90 degrees, and so the resulting angle is acute.
  • If one or both angles are significantly larger but still less than 90 degrees, their sum might exceed 90 degrees. In such cases, the resulting angle could be obtuse (more than 90 degrees but less than 180 degrees).
  • If the sum exactly hits 90 degrees, then the angle is termed a right angle.

Visualizing angles and their combinations can build an intuitive understanding of angles. Consider using a protractor to measure, verify, and visualize each scenario.
Obtuse Angle
An obtuse angle is an angle that is greater than 90 degrees but less than 180 degrees. This is an important category of angle since it encompasses a wide range of possible measurements. Obtuse angles are larger than right angles but smaller than a straight line.

Think about the following scenarios with acute angles:
  • If you have one acute angle larger than 45 degrees and you add it to another acute angle, the resulting angle can become obtuse.
  • For example, adding a 60-degree angle to a 40-degree angle gives you a 100-degree angle, which is obtuse.

Visual aids like drawing these angles and combining them help solidify the understanding of how these angle types relate. As you work with angles, always remember that an obtuse angle cannot exceed 179 degrees.
Right Angle
The right angle is perhaps one of the most recognized angle types, easily identifiable by its 90-degree measure. Right angles are foundational in geometry, forming the basis for squares and rectangles and commonly seen in many structures and everyday objects.

When considering the sum of two acute angles:
  • If each acute angle measures 45 degrees, their sum will be exactly 90 degrees, i.e., a right angle.
  • This scenario is straightforward, where both individual angles are exactly half of a right angle.

Practicing with right angles, such as using a carpenter's square, can give a better sense of what 90 degrees looks and feels like. Understanding how acute angles combine to form a right angle enhances both analytical and practical geometry skills.

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Most popular questions from this chapter

Draw a number line and label points \(\mathrm{F}, \mathrm{G}, \mathrm{H},\) and \(\mathrm{J}\) with the coordinates \(-4 \frac{2}{3}, 2,5,\) and 3.5 respectively. One of these points is the midpoint (the halfway point) between two others. Which is it?

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Q is between \(P\) and \(R\) on a number line. \(P=-8,\) and \(R=4\) a What do we know about the coordinate of Q? b What do we know about the length \(P Q+Q R ?\)

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