Question

given: ( mangle a + mangle b = mangle b + mangle c )
prove: ( mangle c = mangle a )
complete the paragraph proof.
we are given that the sum of the measures of angles a and b is equal to the sum of the measures of angles b and c. the measure of angle b is equal to itself by the dropdown property, so you can subtract that measure from both sides of the equation. now the dropdown by the dropdown property, this means the measure of angle c equals the measure of angle a.
dropdown options: transitive, symmetric, reflexive

Explanation:

1. The reflexive property states that a quantity is equal to itself (e.g., \( m\angle B = m\angle B \)). So the first blank (for the property of \( m\angle B = m\angle B \)) is "reflexive".
2. After subtracting \( m\angle B \) from both sides of \( m\angle A + m\angle B = m\angle B + m\angle C \), we use the subtraction property of equality (or the symmetric property isn't correct here; actually, after subtraction, we get \( m\angle A = m\angle C \), and by the symmetric property of equality (if we consider swapping sides) or more accurately, the transitive or substitution, but the key here is the first property for \( m\angle B = m\angle B \) is reflexive. Then, after subtracting \( m\angle B \) from both sides, we have \( m\angle A = m\angle C \), and by the symmetric property (which states if \( a = b \), then \( b = a \)), we get \( m\angle C = m\angle A \). Wait, no—actually, after subtraction, we get \( m\angle A = m\angle C \), and then by the symmetric property of equality, we can say \( m\angle C = m\angle A \). But the first property (for \( m\angle B = m\angle B \)) is reflexive. So the first dropdown (the one with reflexive, symmetric, transitive) answer is "reflexive", and the second property (to get \( m\angle C = m\angle A \) from \( m\angle A = m\angle C \)) is "symmetric". Wait, let's re - examine:

Given \( m\angle A + m\angle B=m\angle B + m\angle C \). The measure of angle B is equal to itself by the reflexive property (since reflexive property is \( x = x \)). Then, we subtract \( m\angle B \) from both sides (subtraction property of equality) to get \( m\angle A=m\angle C \). Then, by the symmetric property of equality (which says if \( a = b \), then \( b = a \)), we can conclude \( m\angle C = m\angle A \).

So the first blank (the property for \( m\angle B=m\angle B \)) is "reflexive", and the second property (to go from \( m\angle A = m\angle C \) to \( m\angle C = m\angle A \)) is "symmetric".

Answer:

First dropdown: reflexive
Second dropdown: symmetric