https://file.aiquickdraw.com/mathbot/file/1726680196719qv1oo6rk.jpg

Let's analyze the given problem step by step.

We are told that triangles \( \triangle ABD \cong \triangle CBD \), meaning they are congruent. This gives us the following key information:
1. \( \angle ADB = \angle CDB \).
2. \( AB = BC \).
3. \( BD = BD \) (shared side).

From the image, we know:
- \( \angle ADB = 49^\circ \),
- \( \angle CDB = 7x \).

### Step 1: Angle Properties in Triangles
In each triangle, the sum of the angles must equal \(180^\circ\). 

Since \( \triangle ABD \) and \( \triangle CBD \) are congruent, we can apply the following:

For \( \triangle ABD \):
\[
\angle ADB + \angle ABD + \angle DAB = 180^\circ.
\]
Similarly, for \( \triangle CBD \):
\[
\angle CDB + \angle CBD + \angle DCB = 180^\circ.
\]

### Step 2: Angle Relationships
Because the triangles are congruent, we know:
\[
\angle ABD = \angle CBD \quad \text{and} \quad \angle DAB = \angle DCB.
\]

Also, \( \angle ADB = \angle CDB \), so:
\[
49^\circ = 7x.
\]

### Step 3: Solve for \( x \)
We can now solve for \( x \):
\[
7x = 49^\circ,
\]
\[
x = \frac{49^\circ}{7} = 7^\circ.
\]

### Conclusion
Thus, \( x = 7^\circ \).

Would you like more details or have any further questions?

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Here are 5 related questions to extend this topic:
1. How do we prove congruence between two triangles?
2. What are the criteria for triangle congruence (e.g., SSS, SAS, ASA)?
3. How do we calculate unknown angles in a triangle if only two angles are given?
4. How can congruence help us solve problems involving parallel lines and transversals?
5. What are the properties of isosceles triangles, and how do they relate to this problem?

**Tip**: In congruent triangles, corresponding sides and angles are always equal!

If ΔABD ≅ ΔCBD, ∠ADB = 49° and ∠CDB = 7x, what is the value of x?

In this problem, we explore how congruent triangles help us find unknown angles. Learn how to calculate x when ΔABD ≅ ΔCBD, and angle ADB is 49°, while angle CDB is 7x. Suitable for students in Grades 8-10.

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