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Lines $x$ and $y$ intersect to make two pairs of vertical angles, $q, s$ and $r, t$. Fill in the blank space in the given proof to prove $\angle q \cong \angle s$.
|Statements|Reasons|
|---|---|
| $m \angle q + m \angle r=180^{\circ} ...( i )$ | $\angle q$ and $\angle r$ are supplementary |
| $m \angle r+ m \angle s =180^{\circ} ...( i )$ | $\angle r$ and $\angle s$ are supplementary |
| $m \angle q + m \angle r= m \angle r+ m \angle s$ | Algebraic substitution |
| $m \angle q = m \angle s$ | Subtraction property of equality |
|$\angle q \cong s$ | When two angles have the same measure, they're congruent.|
A) Subtraction property of equality
B) Addition property of equality
C) Substitution property
D) Transitive property
Answer (2)
The proof starts with the fact that ∠ q and ∠ r are supplementary, and ∠ r and ∠ s are supplementary.
It uses algebraic substitution to show that m ∠ q + m ∠ r = m ∠ r + m ∠ s .
It applies the subtraction property of equality to deduce that m ∠ q = m ∠ s .
Concludes that A is the correct answer.
Explanation
Analyze the problem and given data The problem provides a proof demonstrating that vertical angles are congruent. We are given that lines x and y intersect, forming vertical angles q , s and r , t . The proof uses the fact that supplementary angles add up to 18 0 ∘ and applies algebraic substitution. The goal is to identify the correct reason for the step where m ∠ q = m ∠ s .
Identify the algebraic operation The step m ∠ q = m ∠ s is derived from the equation m ∠ q + m ∠ r = m ∠ r + m ∠ s . To get from the first equation to the second, we subtract m ∠ r from both sides of the equation:
m ∠ q + m ∠ r − m ∠ r = m ∠ r + m ∠ s − m ∠ r
m ∠ q = m ∠ s
State the property used This operation is justified by the subtraction property of equality, which states that if you subtract the same quantity from both sides of an equation, the equation remains true.
Conclusion Therefore, the correct answer is A) Subtraction property of equality.
Examples
Understanding vertical angles and their properties is crucial in various real-world applications, such as architecture and construction. For example, when designing a building, architects need to ensure that the angles formed by intersecting walls are precise to maintain structural integrity and aesthetic appeal. The property that vertical angles are congruent helps in accurately calculating and verifying these angles, ensuring that the building is constructed according to the design specifications.
To prove that angles q and s are congruent, we demonstrate that m ∠ q = m ∠ s using the subtraction property of equality after establishing their relationships with supplementary angles. Hence, the correct option is A) Subtraction property of equality.
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