Question

hj || qs and tv || hj. complete the proof that m∠put + m∠qrw = 180°.
statement reason
1 hj || qs given
2 tv || hj given
3 ∠put ≅ ∠hip
4 m∠hip + m∠qrw = 180° alternate exterior angles theorem alternate interior angles theorem
5 m∠put + m∠qrw = 180° converse of alternate exterior angles theorem converse of alternate interior angles theorem converse of corresponding angles theorem converse of same - side exterior angles theorem converse of same - side interior angles theorem corresponding angles theorem same - side exterior angles theorem same - side interior angles theorem

Explanation:

Step1: Identify equal - angled pairs

Since $\overleftrightarrow{HJ}\parallel\overleftrightarrow{QS}$ and $\overleftrightarrow{TV}\parallel\overleftrightarrow{HJ}$, by the Alternate Exterior Angles Theorem, $\angle PUT\cong\angle HIP$ because they are alternate exterior angles for parallel lines $\overleftrightarrow{TV}$ and $\overleftrightarrow{HJ}$ with transversal $\overleftrightarrow{PH}$.

Step2: Use angle - sum property

$\angle HIP$ and $\angle QRW$ are same - side exterior angles with respect to parallel lines $\overleftrightarrow{HJ}$ and $\overleftrightarrow{QS}$ and transversal $\overleftrightarrow{PR}$. By the Same - Side Exterior Angles Theorem, $m\angle HIP + m\angle QRW=180^{\circ}$.

Step3: Substitute equal angles

Since $\angle PUT\cong\angle HIP$, we can substitute $\angle HIP$ with $\angle PUT$ in the equation $m\angle HIP + m\angle QRW = 180^{\circ}$. So $m\angle PUT + m\angle QRW=180^{\circ}$.

Answer:

Same - Side Exterior Angles Theorem