Question

proving the congruent supplements theorem
given: $angle 1$ and $angle 2$ are supplements, and $angle 3$ and $angle 2$ are supplements.
prove: $angle 1 cong angle 3$
complete the missing parts of the paragraph proof.
by the definition of $\boldsymbol{square}$ angles, the sum of the measures of angles 1 and 2 is 180 degrees.
likewise, the sum of the measures of angles $\boldsymbol{square}$ is 180 degrees. by the $\boldsymbol{square}$ property, $mangle 1 + mangle 2 = mangle 3 + mangle 2$. subtract the measure of angle $\boldsymbol{square}$ from each side. you get $mangle 1 = mangle 3$, or $angle 1 cong angle 3$, by the definition of congruence.

Explanation:

1. The first blank refers to the type of angles given (supplementary), so we use the definition of supplementary angles.
2. The second blank follows the same logic as the first pair: since ∠3 and ∠2 are supplements, we reference angles 3 and 2.
3. The third blank uses the transitive property of equality, as two quantities equal to 180° are equal to each other.
4. The fourth blank subtracts the common angle (∠2) from both sides to isolate the measures of ∠1 and ∠3.

Answer:

1. First blank: supplementary
2. Second blank: 3 and 2
3. Third blank: transitive (equality)
4. Fourth blank: 2

Full completed proof:
By the definition of supplementary angles, the sum of the measures of angles 1 and 2 is 180 degrees.
Likewise, the sum of the measures of angles 3 and 2 is 180 degrees. By the transitive (equality) property, $m\angle1 + m\angle2 = m\angle3 + m\angle2$. Subtract the measure of angle 2 from each side. You get $m\angle1 = m\angle3$, or $\angle1 \cong \angle3$, by the definition of congruence.