Question

given: m∠1 + m∠3 = 180 prove: t || r 1. click here to insert 1. given 2. click here to insert 2. pairs of vertical angles are congruent. 3. click here to insert 3. substitution property 4. ∠2 and ∠3 are supplementary. 4. click here to insert

Explanation:

Step1: State the given

Given $m\angle1 + m\angle3 = 180$.

Step2: Identify vertical angles

Vertical angles $\angle1$ and $\angle2$ are congruent, so $\angle1\cong\angle2$.

Step3: Apply substitution

Since $\angle1\cong\angle2$, we substitute $\angle1$ with $\angle2$ in $m\angle1 + m\angle3=180$ to get $m\angle2 + m\angle3 = 180$.

Step4: Define supplementary angles

Two angles whose sum is 180 degrees are supplementary, so $\angle2$ and $\angle3$ are supplementary.

Step5: Use parallel - line theorem

If same - side interior angles (in this case $\angle2$ and $\angle3$) are supplementary when two lines are cut by a transversal, then the two lines $t$ and $r$ are parallel.

Answer:

1. $m\angle1 + m\angle3=180$ (Given)
2. $\angle1\cong\angle2$ (Pairs of vertical angles are congruent)
3. $m\angle2 + m\angle3 = 180$ (Substitution Property)
4. $\angle2$ and $\angle3$ are supplementary. (Definition of supplementary angles)
5. $t\parallel r$ (If two lines are cut by a transversal such that a pair of same - side interior angles are supplementary, then the two lines are parallel)