Given: [tex]$m \angle ADE =60^{\circ}$[/tex] and [tex]$m \angle CDF =(3 x+15)^{\circ}$[/tex]
Prove: [tex]$x=15$[/tex]
What is the missing statement and the missing reason in step 5?
Statements | Reasons
--- | ---
1. [tex]$m \angle A D E=60^{\circ}$[/tex] | 1. Given
2. [tex]$m \angle C D F=(3 x+15)^{\circ}$[/tex] | 2. Given
3. [tex]$\angle A D E$[/tex] and [tex]$\angle C D F$[/tex] are vert. [tex]$\angle s$[/tex] | 3. def. of vert. [tex]$\angle s$[/tex]
4. [tex]$\angle A D E=\angle C D F$[/tex] | 4. vert. [tex]$\angle s a$[/tex]
5. [tex]$\angle A D E=m \angle C D F$[/tex] | 5. def. of a
6. | 6. ?
7. [tex]$45=3 x$[/tex] | 7. subtr. prop.
8. [tex]$15=x$[/tex] | 8. div. prop.
Statement: [tex]$60=3 x+15$[/tex]; Reason: substitution
Statement: [tex]$x=15$[/tex]; Reason: subtraction property of equality
Statement: [tex]$60=3 x+15$[/tex]; Reason: transitive property
Statement: [tex]$x=15$[/tex]; Reason: subtraction and division properties of equality
Answer (2)
Recognize that vertical angles are congruent.
Substitute the given angle measures into an equation: 60 = 3 x + 15 .
Solve for x by subtracting 15 from both sides and then dividing by 3.
Conclude that x = 15 , with the missing statement being 60 = 3 x + 15 and the reason being substitution. x = 15
Explanation
Analyze the problem and given data. We are given that m ∠ A D E = 6 0 ∘ and m ∠ C D F = ( 3 x + 15 ) ∘ . We are also given that ∠ A D E and ∠ C D F are vertical angles. Vertical angles are congruent, which means they have the same measure. Therefore, m ∠ A D E = m ∠ C D F .
Apply substitution. Since m ∠ A D E = m ∠ C D F , we can substitute the given values to get 60 = 3 x + 15 . This is because both angles are equal, so their measures must be equal as well.
Solve for x. Now, we solve for x . First, subtract 15 from both sides of the equation:
60 − 15 = 3 x + 15 − 15
45 = 3 x
Find the value of x. Next, divide both sides by 3:
3 45 = 3 3 x
15 = x
So, x = 15 .
Identify the missing statement and reason. The missing statement is 60 = 3 x + 15 , and the missing reason is substitution.
Examples
Understanding angle relationships and solving for unknowns is crucial in various fields, such as architecture and engineering. For instance, when designing a bridge, engineers need to calculate angles to ensure structural stability. If two supporting beams form vertical angles, and one angle is known, the other can be easily determined using the properties of vertical angles and algebraic substitution, ensuring the bridge is built according to precise specifications.
To find x , we start with the equations for vertical angles, giving us 60 = 3 x + 15 . We then solve for x by isolating it, resulting in x = 15 . The missing statement is 60 = 3 x + 15 and the reason is substitution.
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