Question

consider the lines and angles shown in the diagram. which statement is true if and only if line l is perpendicular to line m? a. (mangle1 = mangle4) b. (mangle1 = mangle2) c. (mangle3 = 180^{circ}-mangle4)

Explanation:

Step1: Recall vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle4$ are vertical angles, so $m\angle1 = m\angle4$ always holds, not just when $l\perp m$.

Step2: Recall linear - pair and perpendicular - angle property

When two lines $l$ and $m$ are perpendicular, the four angles formed are right - angles, each measuring $90^{\circ}$. $\angle1$ and $\angle2$ form a linear pair. If $l\perp m$, then $m\angle1=m\angle2 = 90^{\circ}$.

Step3: Recall linear - pair property

$\angle3$ and $\angle4$ always form a linear pair, so $m\angle3=180^{\circ}-m\angle4$ always holds, not just when $l\perp m$.

Answer:

B. $m\angle1 = m\angle2$