4 - 3 study guide and intervention (continued)
congruent triangles
prove triangles congruent two triangles are congruent if and only if their corresponding par
corresponding parts include corresponding angles and corresponding sides. the phrase “if and onl
both the conditional and its converse are true. so, if two polygons are congruent, then their corres
congruent. for triangles, we say, “corresponding parts of congruent triangles are congruent,” or c
example: write a two - column proof.
given: $ \overline { a b } \cong \overline { c b } $, $ \overline { a d } \cong \overline { c d } $, $ \angle b a d \cong \angle b c d $,
$ \overline { b d } $ bisects $ \angle a b c $
prove: $ \triangle a b d \cong \triangle c b d $
proof:
\begin{tabular}{l|l}
statement & reason \\
\hline
1. $ \overline { a b } \cong \overline { c b } $, $ \overline { a d } \cong \overline { c d } $ & 1. given \\
2. $ \overline { b d } \cong \overline { b d } $ & 2. reflexive property of congruence \\
3. $ \angle b a d \cong \angle b c d $ & 3. given \\
4. $ \angle a b d \cong \angle c b d $ & 4. definition of angle bisector \\
5. $ \angle b d a \cong \angle b d c $ & 5. third angles theorem \\
6. $ \triangle a b d \cong \triangle c b d $ & 6. def. of congruent polygons \\
\end{tabular}
exercises
write a two - column proof.
1. given: $ \angle a \cong \angle c $, $ \angle d \cong \angle b $, $ \overline { a d } \cong \overline { c b } $, $ \overline { a e } \cong \overline { c e } $,
$ \overline { a c } $ bisects $ \overline { b d } $
prove: $ \triangle a e d \cong \triangle c e b $
write a paragraph proof.
2. given: $ \overline { b d } $ bisects $ \angle a b c $ and $ \angle a d c $,
$ \overline { a b } \cong \overline { c b } $, $ \overline { a b } \cong \overline { a d } $, $ \overline { c b } \cong \overline { d c } $
prove: $ \triangle a b d \cong \triangle c b d $

Question

Accepted Answer

# Explanation:
## Step1: State the given information
Given: $\angle A\cong\angle C$, $\angle D\cong\angle B$, $\overline{AD}\cong\overline{CB}$, $\overline{AE}\cong\overline{CE}$, $\overline{AC}$ bisects $\overline{BD}$.
## Step2: Use the definition of bisector
Since $\overline{AC}$ bisects $\overline{BD}$, then $\overline{DE}\cong\overline{BE}$.
## Step3: Apply the ASA (Angle - Side - Angle) congruence criterion
In $	riangle AED$ and $	riangle CEB$:
- $\angle A\cong\angle C$ (given)
- $\overline{AD}\cong\overline{CB}$ (given)
- $\angle D\cong\angle B$ (given)

By the ASA congruence criterion, $	riangle AED\cong	riangle CEB$.

# Answer:
The two - column proof for $	riangle AED\cong	riangle CEB$ is as follows:
| Statement | Reason |
| --- | --- |
| 1. $\angle A\cong\angle C$, $\angle D\cong\angle B$, $\overline{AD}\cong\overline{CB}$, $\overline{AE}\cong\overline{CE}$, $\overline{AC}$ bisects $\overline{BD}$ | Given |
| 2. $\overline{DE}\cong\overline{BE}$ | Definition of bisector |
| 3. $	riangle AED\cong	riangle CEB$ | ASA ($\angle A\cong\angle C$, $\overline{AD}\cong\overline{CB}$, $\angle D\cong\angle B$) |

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QUESTION IMAGE

Question

Explanation:

Step1: State the given information

Step2: Use the definition of bisector

Step3: Apply the ASA (Angle - Side - Angle) congruence criterion

Answer:

Statement	Reason
2. $\overline{DE}\cong\overline{BE}$	Definition of bisector
3. $\triangle AED\cong\triangle CEB$	ASA ($\angle A\cong\angle C$, $\overline{AD}\cong\overline{CB}$, $\angle D\cong\angle B$)