Question

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 angles, the sum of the measures of angles 1 and 2 is 180 degrees.
likewise, the sum of the measures of angles is 180 degrees. by the property, ( mangle1 + mangle2 = mangle3 + mangle2 ). subtract the measure of angle from each side. you get ( mangle1 = mangle3 ), or ( angle1 ≅ angle3 ), by the definition of congruence.

Explanation:

• Supplementary angles: By definition, supplementary angles sum to \(180^{\circ}\). So, for \(\angle1\) and \(\angle2\) (supplements) and \(\angle3\) and \(\angle2\) (supplements), their angle - sum equations are based on this definition.
• Substitution property: Since both \(m\angle1 + m\angle2=180\) and \(m\angle3 + m\angle2 = 180\), we can substitute \(180\) in the first equation with \(m\angle3 + m\angle2\) (because they are both equal to \(180\)) to get \(m\angle1 + m\angle2=m\angle3 + m\angle2\).
• Subtraction of \(\angle2\): To isolate \(m\angle1\) and \(m\angle3\), we use the subtraction property of equality. If \(a + c=b + c\), then \(a=b\) (here \(a = m\angle1\), \(b=m\angle3\), and \(c = m\angle2\)).

Answer:

1. supplementary
2. \(3\) and \(2\)
3. substitution
4. \(2\)