triangle ace is shown, with b the midpoint of $\overline{ac}$ and d the midpoint of $\overline{ce}$.
move symbols into the table to prove that $\overline{ae} \parallel \overline{bd}$.
| statements | reasons |
| --- | --- |
| b is the midpoint of $\overline{ac}$; d is the midpoint of $\overline{ce}$ | given |
| $\overline{ac} = 2\cdot\underline{quadquad}; \overline{ce} = 2\cdot\underline{quadquad}$ | definition of midpoint |
| $\underline{quadquad} \cong \underline{quadquad}$ | reflexive property |
| $\triangle ace \sim \triangle bcd$ | sas similarity |
| $\underline{quadquad} \cong \underline{quadquad}$ | corresponding angles of similar triangles are congruent. |
| $\overline{ae} \parallel \overline{bd}$ | if corresponding angles are congruent, then the lines are parallel. |
symbols to choose from: $\overline{ab}$, $\overline{bc}$, $\overline{cd}$, $\overline{de}$, $\angle a$, $\angle c$, $\angle e$, $\angle cdb$, $\angle dbc$

Question

Accepted Answer

# Explanation:
## Step1: Fill midpoint segments
$\overline{AC} = 2\cdot\overline{BC}$; $\overline{CE} = 2\cdot\overline{CD}$
## Step2: Reflexive congruent angle
$\angle C \cong \angle C$
## Step3: Corresponding congruent angles
$\angle A \cong \angle DBC$ (or $\angle E \cong \angle CDB$)

# Answer:
| Statements | Reasons |
|------------|---------|
| $B$ is the midpoint of $\overline{AC}$; $D$ is the midpoint of $\overline{CE}$ | Given |
| $\overline{AC} = 2\cdot\overline{BC}$; $\overline{CE} = 2\cdot\overline{CD}$ | Definition of midpoint |
| $\angle C \cong \angle C$ | Reflexive property |
| $	riangle ACE \sim 	riangle BCD$ | SAS similarity |
| $\angle A \cong \angle DBC$ | Corresponding angles of similar triangles are congruent. |
| $\overline{AE} \parallel \overline{BD}$ | If corresponding angles are congruent, then the lines are parallel. |

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QUESTION IMAGE

Question

Explanation:

Step1: Fill midpoint segments

Step2: Reflexive congruent angle

Step3: Corresponding congruent angles

Answer:

statements	reasons
\\(\overline{ac} = 2\cdot\underline{quadquad}; \overline{ce} = 2\cdot\underline{quadquad}\\)	definition of midpoint
\\(\underline{quadquad} \cong \underline{quadquad}\\)	reflexive property
\\(\triangle ace \sim \triangle bcd\\)	sas similarity
\\(\underline{quadquad} \cong \underline{quadquad}\\)	corresponding angles of similar triangles are congruent.
\\(\overline{ae} \parallel \overline{bd}\\)	if corresponding angles are congruent, then the lines are parallel.

Statements	Reasons
$\overline{AC} = 2\cdot\overline{BC}$; $\overline{CE} = 2\cdot\overline{CD}$	Definition of midpoint
$\angle C \cong \angle C$	Reflexive property
$\triangle ACE \sim \triangle BCD$	SAS similarity
$\angle A \cong \angle DBC$	Corresponding angles of similar triangles are congruent.
$\overline{AE} \parallel \overline{BD}$	If corresponding angles are congruent, then the lines are parallel.