1. Find the value of $x$ that makes $m \parallel n$. **a.** Given angles: $100^\circ$ and $2x^\circ$ are corresponding angles. Since $m \parallel n$, corresponding angles are equal: $$100 = 2x$$ Solve for $x$: $$x = \frac{100}{2} = 50$$ **b.** Given angles: $(5x + 23)^\circ$ and $(7x + 13)^\circ$ are alternate interior angles. Since $m \parallel n$, alternate interior angles are equal: $$5x + 23 = 7x + 13$$ Solve for $x$: $$23 - 13 = 7x - 5x$$ $$10 = 2x$$ $$x = 5$$ 2. Draw two parallel lines and a transversal cutting through at $35^\circ$. - Corresponding angles are equal to $35^\circ$. - Alternate interior angles are also $35^\circ$. - Consecutive interior angles are supplementary, so $180 - 35 = 145^\circ$. Label all angles accordingly: - Angles corresponding to $35^\circ$ are $35^\circ$. - Angles supplementary to $35^\circ$ are $145^\circ$. 3. Proof that $p \parallel r$ given $g \parallel h$ and $\angle 1 \cong \angle 2$. **Given:** $g \parallel h$, $\angle 1 \cong \angle 2$ **To Prove:** $p \parallel r$ **Proof (Two-column):** | Statements | Reasons | |---|---| | 1. $g \parallel h$ | Given | | 2. $\angle 1 \cong \angle 2$ | Given | | 3. $\angle 1$ and $\angle 3$ are alternate interior angles formed by transversal $g$ with lines $p$ and $r$ | Definition of alternate interior angles | | 4. $\angle 2$ and $\angle 3$ are corresponding angles | Angle relationships in the diagram | | 5. $\angle 1 \cong \angle 2$ implies $\angle 3$ is congruent to $\angle 1$ and $\angle 2$ | Transitive property of congruence | | 6. Since $\angle 1 \cong \angle 3$, lines $p$ and $r$ are parallel | Converse of alternate interior angles theorem | Thus, $p \parallel r$ is proven.