Question

complementary and supplementary angles
assignment active
proving the congruent supplements theorem
given: ∠1 and ∠2 are supplements, and ∠3 and ∠2 are supplements.
prove: ∠1 ≅ ∠3
complete the missing parts of the paragraph proof.
by the definition of $\boldsymbol{hspace{5em}}$ angles, the sum of the measures of angles 1 and 2 is 180 degrees.
likewise, the sum of the measures of angles $\boldsymbol{hspace{5em}}$ is 180 degrees. by the $\boldsymbol{hspace{5em}}$ property, $m∠1 + m∠2 = m∠3 + m∠2$. subtract the measure of angle $\boldsymbol{hspace{2em}}$ from each side. you get $m∠1 = m∠3$, or $∠1 ≅ ∠3$, by the definition of congruence.

Explanation:

1. The first blank refers to the type of angles whose sum is 180 degrees, which are supplementary angles, matching the given information.
2. The second blank refers to the other pair of supplementary angles given: ∠3 and ∠2.
3. The third blank refers to the property that if two expressions both equal 180, they equal each other, which is the transitive property of equality.
4. The fourth blank is the angle measure we subtract from both sides to isolate \(m\angle1\) and \(m\angle3\), which is \(m\angle2\).

Answer:

1. supplementary
2. 3 and 2
3. transitive (equality)
4. 2

Filled proof for context:
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 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.