Question

if ( mangle a = mangle b ) and ( mangle a + mangle c = mangle d ), then ( mangle b + mangle c = mangle d ). which property is shown?
reflexive property
substitution property
symmetric property
transitive property

Explanation:

1. Recall the definitions of each property:

• Reflexive property: A quantity is equal to itself (e.g., \(a = a\)). This doesn't match the given situation.
• Substitution property: If \(a = b\), then \(a\) can be replaced by \(b\) (or vice versa) in any equation. Here, since \(m\angle A=m\angle B\), we substitute \(m\angle A\) with \(m\angle B\) in the equation \(m\angle A + m\angle C=m\angle D\) to get \(m\angle B + m\angle C=m\angle D\).
• Symmetric property: If \(a = b\), then \(b = a\). This is about reversing the equality, not substituting.
• Transitive property: If \(a = b\) and \(b = c\), then \(a = c\). This is about a chain of equalities, not substitution.

1. Based on the above analysis, the substitution property is used here as we substitute \(m\angle A\) with \(m\angle B\) in the second equation.

Answer:

substitution property (the option corresponding to "substitution property")