Question

writing a paragraph proof
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 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

Explanation:

Step1: Analyze the given equation

We start with the given equation \( m\angle A + m\angle B = m\angle B + m\angle C \).

Step2: Apply the Subtraction Property of Equality

Subtract \( m\angle B \) from both sides of the equation. By the Subtraction Property of Equality, if \( a + b = b + c \), then subtracting \( b \) from both sides gives \( a = c \). So, subtracting \( m\angle B \) from both sides of \( m\angle A + m\angle B = m\angle B + m\angle C \) gives \( m\angle A = m\angle C \) (which is equivalent to \( m\angle C = m\angle A \) by the Symmetric Property of Equality, but first we get \( m\angle A = m\angle C \) after subtraction). The first blank (the blue one) should be "measure of angle A equals the measure of angle C" because after subtracting \( m\angle B \) from both sides, we have \( m\angle A = m\angle C \). The second property is the Symmetric Property of Equality (since we can rewrite \( m\angle A = m\angle C \) as \( m\angle C = m\angle A \)) or we can think of the result after subtraction as directly giving \( m\angle A = m\angle C \), and then the property that allows us to say \( m\angle C = m\angle A \) from \( m\angle A = m\angle C \) is the Symmetric Property of Equality. But for the first blank, the correct option is "measure of angle A equals the measure of angle C".

Answer:

First blank: measure of angle A equals the measure of angle C; Second property: Symmetric (or the property that if \( a = b \) then \( b = a \), which is the Symmetric Property of Equality)