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 reflexive property, so you can subtract that measure from both sides of the equation. now the dropdown with options: measure of angle a equals the measure of angle c, measure of angle a equals the measure of angle b, measure of angle b equals the measure of angle c by the dropdown property, this means the measure of angle c equals the measure of angle a.

Explanation:

Step1: Analyze the given equation

We have \( m\angle A + m\angle B = m\angle B + m\angle C \).

Step2: Subtract \( m\angle B \) from both sides

Using the subtraction property of equality (subtracting the same quantity from both sides of an equation preserves equality), we subtract \( m\angle B \) from both sides:
\( m\angle A + m\angle B - m\angle B = m\angle B + m\angle C - m\angle B \)
Simplifying both sides, \( m\angle A = m\angle C \).
The first blank (after subtraction) should be "measure of angle A equals the measure of angle C", and the second property is the "subtraction" (or "symmetric" in the context of equality, but here subtraction leads to \( m\angle A = m\angle C \), and then by symmetric property of equality, \( m\angle C = m\angle A \)). But from the options, the first part is "measure of angle A equals the measure of angle C", and the property is "symmetric" (since if \( a = b \), then \( b = a \)) or "subtraction" leading to \( a = b \), then symmetric gives \( b = a \). But the first dropdown after subtraction: the result is \( m\angle A = m\angle C \), so the first option in the box is "measure of angle A equals the measure of angle C". Then the property for \( m\angle C = m\angle A \) from \( m\angle A = m\angle C \) is the symmetric property of equality.

Answer:

First blank: measure of angle A equals the measure of angle C
Second property: symmetric (or subtraction followed by symmetric, but the key first part is the angle A = angle C statement)

(For the first dropdown in the box: select "measure of angle A equals the measure of angle C"; for the second dropdown: "symmetric" property of equality)