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Magazine Chapter 1: Problem 28
In the proof to the right, provide the missing reasons.
Given: \(\angle 1\) and \(\angle 2\) are complementary \(\angle 1\) is acute
Prove: \(\angle 2\) is also acute
Statements
1\. \(\angle 1\) and \(\angle 2\) are
complementary
2\. \(\mathrm{m} \angle 1+\mathrm{m} \angle 2=90\)
3\. \(\angle 1\) is acute
4\. Where \(m \angle 1=x\)
\(0
Short Answer
Expert verified
Using definitions and algebra, we show \( \angle 2 \) is less than 90掳, making it acute.
Step by step solution
01
Identify Complementary Angles
The given information states that \( \angle 1 \) and \( \angle 2 \) are complementary angles. By definition, complementary angles are two angles whose sum is 90 degrees. Hence, the reason for statement 1 is 'Definition of Complementary Angles.'
02
Sum of Angles Equation
From the definition of complementary angles, we can write the equation for the sum of \( \angle 1 \) and \( \angle 2 \). So, the reason for statement 2 is 'Definition of Complementary Angles.'
03
Given Angle is Acute
The problem states that \( \angle 1 \) is an acute angle, which means its measure is less than 90 degrees. Hence, for statement 3, the reason is 'Given.'
04
Define Variable for Measure
To simplify calculations, the measure of \( \angle 1 \) is represented as \( x \) where \( 0 < x < 90 \). The reason for statement 4 is 'Definition of an Acute Angle.'
05
Substitute Variable in Equation
Substituting \( x \) for the measure of \( \angle 1 \) in the equation derived from complementary angles, we get \( x + \text{m} \angle 2 = 90 \). Therefore, the reason for statement 5 is 'Substitution.'
06
Solve for Measure of Angle 2
Rearranging the equation from step 5 to isolate \( \text{m} \angle 2 \), we get \( \text{m} \angle 2 = 90 - x \). The reason for statement 6 is 'Algebraic Manipulation.'
07
Solve Inequality for Angle Measure
Substituting \( x \) with any value \( 0 < x < 90 \) in \( 90 - x \), we maintain the inequality \(-x < 0 < 90-x \). This implies the reason for statement 7 is 'Properties of Inequality.'
08
Substitute and Simplify Inequality
From statement 7, we need to ensure \( 90 - x < 90 \). This simplifies directly since \( -x < 0 \), giving us 'Substitution of an Acute Angle' for statement 8.
09
Conclude Inequality for Angle Measure
Since \( 0 < 90 - x < 90 \), we establish that the measure of angle 2, \( ext{m} \angle 2 \), satisfies the conditions of being less than 90 but greater than 0. Therefore, the reason for statement 9 is 'Transitive Property of Inequality.'
10
Determine Measure Interval
Combine the elements to clearly state \( 0 < \text{m} \angle 2 < 90 \), identifying it as an acute angle based on its measure, for statement 10 the reason is 'Definition of an Acute Angle.'
11
Conclude Acute Property of Angle 2
From the established inequality, angle 2 is acute since it fits the definition of having a measure between 0 and 90 degrees. Thus, the reason for statement 11 is 'Definition of an Acute Angle.'
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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 two angles whose measures add up to exactly 90 degrees. This is an essential concept in geometry that helps us understand the relationships between different angles.
- Consider two angles: if one angle measures 30 degrees, the complementary angle must measure 60 degrees. Why? Because they must sum up to 90 degrees.
- If two angles are complementary, they can be neatly visualized as forming a right angle when placed adjacent to each other.
- The equation to represent complementary angles is: \( \angle A + \angle B = 90^\circ \).
Acute Angles
Acute angles are angles that measure less than 90 degrees. This definition helps us quickly identify and classify angles based on their size.
- Common examples of acute angles are found in triangles, especially equilateral and right triangles.
- An angle measuring 45 degrees, for example, is acute because it is less than 90 degrees.
- Mathematically, if an angle \( \angle X \) is acute, then \( 0 < \angle X < 90 \).
Algebraic Manipulation
Algebraic manipulation involves using algebra to rearrange, substitute, and solve equations. This technique is widely used in solving geometry proof problems and finding unknowns.
- Start by stating known information in algebraic terms. For example, if \( \angle 1 \) is complementary to \( \angle 2 \), you express it as \( m\angle 1 + m\angle 2 = 90 \).
- Substitute given or expressed values in the equations, such as \( m\angle 1 = x \), then adjust the equation to isolate the unknown angle.
- Rearranging equations can reveal new insights or simplify the problem, such as what is done in step 5 and 6 of our proof.
