Let's restate the proposition for clarity: **Proposition:** If two lines are intersected by a transversal such that a pair of same-side interior angles is supplementary, then the other pair of same-side interior angles is also supplementary. --- Let's proceed step by step. ## **Diagram and Notation** Let lines \( \ell_1 \) and \( \ell_2 \) be intersected by a transversal \( t \). Let the points of intersection be \( A \) (on \( \ell_1 \)) and \( B \) (on \( \ell_2 \)). Let the angles formed be: - At \( A \): angles \( 1 \) and \( 2 \) (on opposite sides of the transversal) - At \( B \): angles \( 3 \) and \( 4 \) (on opposite sides of the transversal) Let us label as follows (as per standard convention): ``` / / 1 / / 3 / / / / -----/------/---- t (transversal) / / 2 / / 4 / / \ell_1 \ell_2 ``` - Angles \( 1 \) and \( 3 \) are interior and on the same side of the transversal (say, left). - Angles \( 2 \) and \( 4 \) are interior and on the other same side (right). ## **Given** A pair of same-side interior angles is supplementary. Let's suppose \( 1 + 3 = 180^\circ \). ## **To Prove** The other pair of same-side interior angles is also supplementary; i.e., \( 2 + 4 = 180^\circ \). --- ## **Proof** ### **Step 1: Use the Linear Pair Relationship** At each intersection, the two angles on a straight line are supplementary (linear pair): - At \( A \): \( 1 + 2 = 180^\circ \) - At \( B \): \( 3 + 4 = 180^\circ \) ### **Step 2: Express \( 2 \) and \( 4 \) in Terms of \( 1 \) and \( 3 \)** From above: - \( 2 = 180^\circ - 1 \) - \( 4 = 180^\circ - 3 \) ### **Step 3: Sum \( 2 + 4 \)** \[ 2 + 4 = (180^\circ - 1) + (180^\circ - 3) = 360^\circ - (1 + 3) \] But we're given \( 1 + 3 = 180^\circ \), so: \[ 2 + 4 = 360^\circ - 180^\circ = 180^\circ \] --- ## **Conclusion** Thus, if one pair of same-side interior angles is supplementary, the other pair is also supplementary. --- ### **Boxed Statement** \[ \boxed{ \text{If two lines are intersected by a transversal such that a pair of same-side interior angles is supplementary, then the other pair is also supplementary.} } \] **Q.E.D.**