Question

consider the figure, where line f is parallel to line g. which statement is true? angles 1, 2, and 3 combine to form a straight line, so m∠1 + m∠2 + m∠3 = 180°. since lines f and g are parallel, m∠1 = m∠5 and m∠3 = m∠6 by the alternate interior angles theorem. therefore, m∠5 + m∠2 + m∠6 = 180° by substitution. angles 1, 2, and 3 combine to form a straight line, so m∠1 + m∠2 + m∠3 = 180°. since lines f and g are parallel, m∠1 = m∠5 and m∠3 = m∠6 by the alternate interior angles theorem. therefore, m∠5 + m∠2 + m∠6 = 180° by subtraction. angles 1, 2, and 3 combine to form a straight line, so m∠1 + m∠2 + m∠3 = 180°. since lines f and g are parallel, m∠1 = m∠5 and m∠3 = m∠6 by the same - side interior angles theorem. therefore, m∠5 + m∠2 + m∠6 = 180° by substitution. angles 1, 2, and 3 combine to form a straight line, so m∠1 + m∠2 + m∠3 = 180°. since lines f and g are parallel, m∠1 = m∠5 and m∠3 = m∠6 by the corresponding angles theorem. therefore, m∠5 + m∠2 + m∠6 = 180° by substitution.

Explanation:

Step1: Recall angle - line relationship

Angles 1, 2, and 3 are adjacent and form a straight - line. So, $m\angle1 + m\angle2 + m\angle3=180^{\circ}$ by the definition of a straight - angle.

Step2: Apply parallel - line theorem

Since lines $f$ and $g$ are parallel, $\angle1$ and $\angle5$ are alternate interior angles, and $\angle3$ and $\angle6$ are alternate interior angles. By the Alternate Interior Angles Theorem, $m\angle1 = m\angle5$ and $m\angle3 = m\angle6$.

Step3: Use substitution

Substitute $m\angle1$ with $m\angle5$ and $m\angle3$ with $m\angle6$ in the equation $m\angle1 + m\angle2 + m\angle3 = 180^{\circ}$. We get $m\angle5 + m\angle2 + m\angle6 = 180^{\circ}$.

Answer:

Angles 1, 2, and 3 combine to form a straight line, so $m\angle1 + m\angle2 + m\angle3 = 180^{\circ}$. Since lines $f$ and $g$ are parallel, $m\angle1 = m\angle5$ and $m\angle3 = m\angle6$ by the Alternate Interior Angles Theorem. Therefore, $m\angle5 + m\angle2 + m\angle6 = 180^{\circ}$ by substitution.