If m∠A = m∠B and m∠A + m∠C = m∠D, then m∠B + m∠C = m∠D. Which property is shown? reflexive property substitution property symmetric property transitive property
The correct answer and explanation is:
Correct Answer: Substitution Property
Explanation:
The Substitution Property in mathematics allows one to replace one quantity with another that is equal to it. In other words, if two values are equal, one can be substituted for the other in any expression or equation without changing the truth of the statement.
Let’s break down the given problem step-by-step:
- We are given that:
- m∠A=m∠Bm∠A = m∠B (Statement 1)
- m∠A+m∠C=m∠Dm∠A + m∠C = m∠D (Statement 2)
- Now, if m∠A=m∠Bm∠A = m∠B, then we can substitute m∠Bm∠B in place of m∠Am∠A in any expression where m∠Am∠A appears.
- Looking at Statement 2:
- m∠A+m∠C=m∠Dm∠A + m∠C = m∠D
- Substituting m∠Bm∠B for m∠Am∠A, based on Statement 1:
- m∠B+m∠C=m∠Dm∠B + m∠C = m∠D
This final equation shows the result of applying the Substitution Property.
Why It’s Not the Other Properties:
- Reflexive Property: States that a quantity is equal to itself (e.g., m∠A=m∠Am∠A = m∠A). This is not what’s happening in the problem.
- Symmetric Property: States that if a=ba = b, then b=ab = a. While related to equality, it doesn’t involve substitution into another equation.
- Transitive Property: States that if a=ba = b and b=cb = c, then a=ca = c. This is not about substitution into an expression, but about equality chaining.
Conclusion:
The correct reasoning behind concluding m∠B+m∠C=m∠Dm∠B + m∠C = m∠D from the given statements is based entirely on replacing one equal angle measure with another. This is a clear example of the Substitution Property.