use the following diagram to prove the perpendicular bisector theorem.
given: $ \overline { c d } $ is the perpendicular bisector of $ \overline { a b } $.
prove: $ \overline { a d } \cong \overline { b d } $
$ \overline { c d } $ is the perpendicular bisector of $ \overline { a b } $.
$ \overline { a c } \cong \overline { b c } $
$ m \angle a c d = 90 ^ { \circ }, m \angle b c d = 90 ^ { \circ } $
$ \overline { c d } \cong \overline { c d } $
$ \delta a c d \cong \delta b c d $

Question

Accepted Answer

# Explanation:
## Step1: Fill in the first blank
The first statement is given, so the reason is "Given".
## Step2: Second statement
Since $CD$ is the perpendicular bisector of $AB$, by the definition of a perpendicular bisector, $C$ is the mid - point of $AB$. So the statement is "$\overline{AC}\cong\overline{BC}$" (mid - point property) and the reason for the second blank (the statement "$\overline{AC}\cong\overline{BC}$") is "Definition of a bisector".
## Step3: Third statement
The angles $\angle ACD$ and $\angle BCD$ are right angles. The reason for  $m\angle ACD = 90^{\circ},m\angle BCD=90^{\circ}$ is "Definition of perpendicular lines" (because $CD\perp AB$).
## Step4: Fourth statement
Since $\angle ACD$ and $\angle BCD$ are right angles, $\angle ACD\cong\angle BCD$ (all right angles are congruent).
## Step5: Fifth statement
For $\triangle ACD$ and $\triangle BCD$, we have $\overline{AC}\cong\overline{BC}$, $\angle ACD\cong\angle BCD$, and $\overline{CD}\cong\overline{CD}$ (reflexive property). By the Side - Angle - Side (SAS) congruence criterion, $\triangle ACD\cong\triangle BCD$.
## Step6: Sixth statement
Since $\triangle ACD\cong\triangle BCD$, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, $\overline{AD}\cong\overline{BD}$.

# Answer:
| Statements | Reasons |
|---|---|
|$\overline{CD}$ is the perpendicular bisector of $\overline{AB}$.| Given |
|$\overline{AC}\cong\overline{BC}$| Definition of a bisector |
|$m\angle ACD = 90^{\circ},m\angle BCD = 90^{\circ}$| Definition of perpendicular lines |
|$\angle ACD\cong\angle BCD$| All right angles are congruent |
|$\overline{CD}\cong\overline{CD}$| Reflexive property |
|$\triangle ACD\cong\triangle BCD$| SAS congruence criterion |
|$\overline{AD}\cong\overline{BD}$| CPCTC |

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Fill in the first blank

Step2: Second statement

Step3: Third statement

Step4: Fourth statement

Step5: Fifth statement

Step6: Sixth statement

Answer:

Statements	Reasons
\(\overline{AC}\cong\overline{BC}\)	Definition of a bisector
\(m\angle ACD = 90^{\circ},m\angle BCD = 90^{\circ}\)	Definition of perpendicular lines
\(\angle ACD\cong\angle BCD\)	All right angles are congruent
\(\overline{CD}\cong\overline{CD}\)	Reflexive property
\(\triangle ACD\cong\triangle BCD\)	SAS congruence criterion
\(\overline{AD}\cong\overline{BD}\)	CPCTC