Question

line $ell$ is the perpendicular bisector of $overline{ac}$, and $d$ is any point on $ell$. complete the proof that a point on the perpendicular bisector of a line segment is equidistant from the segment’s endpoints. hint: we need to show that the distance between $a$ and $d$ is the same as the distance between $c$ and $d$. statement reason 1 $overline{ac}perpoverline{bd}$ definition of perpendicular bisector. 2 $overline{ab}congoverline{cb}$ definition of perpendicular bisector. 3 $angle abd$ & $angle cbd$ are both right angles definition of perpendicular. 4 $overline{bd}congoverline{bd}$ line segments are congruent to themselves. 5 $\triangle abdcong\triangle cbd$ congruency postulate (2, 3, 4). 6 corresponding parts of congruent triangles have the same measure.

Explanation:

Step1: Identify congruency postulate

Since we have $\overline{AB}\cong\overline{CB}$ (Side), $\angle ABD\cong\angle CBD$ (Right - angle, Angle) and $\overline{BD}\cong\overline{BD}$ (Side), by the Side - Angle - Side (SAS) congruency postulate, $\triangle ABD\cong\triangle CBD$.

Step2: Determine the final statement

Since $\triangle ABD\cong\triangle CBD$, by the property that corresponding parts of congruent triangles are congruent (CPCTC), we have $\overline{AD}\cong\overline{CD}$.

Answer:

1. SAS; 6. $\overline{AD}\cong\overline{CD}$