given the diagram as marked, which statement is correct?
options:
- $\triangle aec \cong \triangle adb$ and $ae = ab$
- $\triangle aed \cong \triangle abc$ and $ae = ab$
- $\triangle aed \cong \triangle acb$ and $m\angle ade = m\angle acb$
- $\triangle abd \cong \triangle ace$ and $m\angle e = m\angle b$

Question

Accepted Answer

# Explanation:
## Step1: Analyze given diagram
From the diagram, we see: $ ED = CB $ (marked with single tick), $ AD = AC $ (marked with double ticks), $ \angle EAD=\angle BAC $ (marked with arcs). Also, $ \angle EAD + \angle DAC=\angle BAC+\angle DAC $, so $ \angle EAC=\angle BAD $. And $ ED + DC=CB + DC $, so $ EC = DB $.

## Step2: Check triangle congruence
For $ \triangle ABD $ and $ \triangle ACE $:
- $ AD = AC $ (given)
- $ \angle BAD=\angle CAE $ (proven above)
- $ AB = AE $? Wait, no, wait: Wait, $ ED = CB $, $ AD = AC $, $ \angle EAD=\angle BAC $, also $ EB $ has $ ED = CB $, so $ AE = AB $? Wait, no, let's re - check the last option: $ \triangle ABD\cong\triangle ACE $ by SAS? Wait, $ AD = AC $, $ \angle BAD=\angle CAE $, and $ AB = AE $? Wait, no, wait the sides: $ EB $ is a line with $ ED = CB $, so $ ED + DC=CB + DC\Rightarrow EC = DB $. Also, $ AD = AC $, $ \angle EAC=\angle BAD $ (since $ \angle EAD=\angle BAC $, adding $ \angle DAC $ to both). So in $ \triangle ABD $ and $ \triangle ACE $:
  - $ AD = AC $
  - $ \angle BAD=\angle CAE $
  - $ AB = AE $? Wait, no, maybe $ AE = AB $ because the triangle $ AEB $ has $ ED = CB $, so it's isoceles? Wait, the last option says $ \triangle ABD\cong\triangle ACE $ and $ m\angle E = m\angle B $. Let's check congruence:
    - $ AD = AC $ (given)
    - $ \angle BAD=\angle CAE $ (as $ \angle EAD=\angle BAC $, so $ \angle EAD+\angle DAC=\angle BAC + \angle DAC\Rightarrow\angle EAC=\angle BAD $)
    - $ AB = AE $? Wait, if $ ED = CB $ and $ AD = AC $, and the angles at $ A $ are equal, then by SAS, $ \triangle ABD\cong\triangle ACE $ (since $ AB = AE $ (because the triangle is isoceles? Wait, the marks on $ EB $ show $ ED = CB $, so $ AE = AB $ (since the triangle $ AEB $ has $ ED = CB $, so it's isoceles with $ AE = AB $). Then, in $ \triangle ABD $ and $ \triangle ACE $: $ AB = AE $, $ \angle BAD=\angle CAE $, $ AD = AC $, so by SAS, $ \triangle ABD\cong\triangle ACE $. Then, corresponding angles $ \angle E=\angle B $ (since they are corresponding angles of congruent triangles).

Now check other options:
- First option: $ \triangle AEC\cong\triangle ADB $: Let's see, $ EC = DB $ (as $ ED = CB $, $ EC=ED + DC $, $ DB = DC + CB $), $ AC = AD $, $ \angle EAC=\angle BAD $. So $ \triangle AEC\cong\triangle ADB $ by SAS, but does $ AE = AB $? If $ \triangle AEC\cong\triangle ADB $, then $ AE = AD $? No, $ AE $ should correspond to $ AB $. Wait, no, in $ \triangle AEC $ and $ \triangle ADB $, $ AE $ corresponds to $ AB $, $ EC $ to $ DB $, $ AC $ to $ AD $. So if $ \triangle AEC\cong\triangle ADB $, then $ AE = AB $, but let's check the angle. Wait, the first option says $ \triangle AEC\cong\triangle ADB $ and $ AE = AB $. But let's check the last option again.

Wait, the last option: $ \triangle ABD\cong\triangle ACE $. Let's use SAS: $ AD = AC $, $ \angle BAD=\angle CAE $, $ AB = AE $ (since $ ED = CB $, the triangle $ AEB $ is isoceles with $ AE = AB $). So $ \triangle ABD\cong\triangle ACE $ (SAS). Then, corresponding angles $ \angle E=\angle B $, which matches the last option.

The other options:
- Second option: $ \triangle AED\cong\triangle ABC $: $ ED = CB $, $ AD = AC $, but $ \angle EAD=\angle BAC $, but the sides $ AE $ and $ AB $, $ ED $ and $ BC $, $ AD $ and $ AC $. Wait, $ \triangle AED $ and $ \triangle ABC $: $ ED = BC $, $ AD = AC $, $ \angle EAD=\angle BAC $, but the included angles? No, the angle between $ ED $ and $ AD $ is $ \angle ADE $, and between $ BC $ and $ AC $ is $ \angle ACB $, not the same as $ \angle EAD $ and $ \angle BAC $. So not congruent.
- Third option: $ \triangle AED\cong\triangle ACB $: Similar to second option, not congruent.
- First option: $ \triangle AEC\cong\triangle ADB $, but does $ AE = AB $? If $ \triangle AEC\cong\triangle ADB $, then $ AE = AB $ (corresponding sides), but let's check the angles. Wait, but the last option's congruence $ \triangle ABD\cong\triangle ACE $ gives $ \angle E=\angle B $, which is a more direct correspondence.

# Answer:
$ \triangle ABD \cong \triangle ACE $ and $ m\angle E = m\angle B $ (the last option: $ \triangle ABD \cong \triangle ACE $ and $ m\angle E = m\angle B $)

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze given diagram

Step2: Check triangle congruence

Answer: