Question

jon wants to prove that points on a perpendicular bisector of a line segment are equidistant from the segments endpoints. he draws the diagram below and uses it to construct a proof. which describes a strategy jon can use for constructing a proof? a. he can show that ac = bc by showing that line m bisects ∠acb. b. he can show that line m bisects ∠acb by showing that △acd≅△bcd by sas. c. he can show that ac = bc by showing that △acd≅△bcd by sas. d. he can show that line m bisects ∠acb by showing that ad = bd.

Explanation:

Step1: Recall properties of perpendicular bisector

Line $m$ is perpendicular bisector of $AB$, so $AD = BD$ and $\angle ADC=\angle BDC = 90^{\circ}$, and $CD$ is common to $\triangle ACD$ and $\triangle BCD$.

Step2: Apply SAS - Side - Angle - Side congruence criterion

In $\triangle ACD$ and $\triangle BCD$, we have $AD = BD$, $\angle ADC=\angle BDC$ and $CD=CD$. So, $\triangle ACD\cong\triangle BCD$ by SAS.

Step3: Use congruent - triangle property

If $\triangle ACD\cong\triangle BCD$, then corresponding sides are equal, so $AC = BC$.

Answer:

C. He can show that $AC = BC$ by showing that $\triangle ACD\cong\triangle BCD$ by SAS.