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The image contains a geometric problem in Vietnamese. I will translate the problem into English and then solve it.

### Problem Translation:
Given:
- \( \Delta ABC \), \( D \) is the midpoint of \( AC \).
- \( E \) is the midpoint of \( BC \).
- \( BD \) intersects \( AE \) at point \( O \).

a) Prove that \( \triangle AOB = \triangle COB \).

b) Prove that \( \angle AOB + \angle BOC = 180^\circ \).

### Solution:

#### Part a) Prove that \( \triangle AOB = \triangle COB \).

Since \( D \) and \( E \) are midpoints of \( AC \) and \( BC \) respectively, we have:

1. \( AD = DC \) and \( BE = EC \).
2. \( BD \) and \( AE \) are the medians of \( \triangle ABC \).
3. \( O \) is the centroid of \( \triangle ABC \), meaning \( AO = \frac{2}{3}AE \) and \( BO = \frac{2}{3}BD \).

Now, let's consider the triangles \( \triangle AOB \) and \( \triangle COB \):

- \( AB \) is common to both triangles.
- \( \angle AOB = \angle COB \) since these angles are vertically opposite.
- \( BO = BO \) (common side).

Therefore, by the **Side-Angle-Side (SAS)** criterion, \( \triangle AOB = \triangle COB \).

#### Part b) Prove that \( \angle AOB + \angle BOC = 180^\circ \).

From the result of part a), we know that \( \triangle AOB \) and \( \triangle COB \) are congruent.

Since \( \angle AOB \) and \( \angle BOC \) are vertically opposite angles, they must be equal. Let \( \angle AOB = \theta \).

Thus, \( \angle AOB + \angle BOC = \theta + \theta = 2\theta \).

However, since these angles are on a straight line formed by \( AO \) and \( BO \), we have:

\[ \angle AOB + \angle BOC = 180^\circ \]

Hence, we have proven that \( \angle AOB + \angle BOC = 180^\circ \).

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Do you want details or have any questions? Here are 5 related questions:

1. How can we determine the centroid of a triangle given the coordinates of its vertices?
2. What are the properties of medians in a triangle?
3. Can you explain the Side-Angle-Side (SAS) criterion in detail?
4. How does the centroid of a triangle divide the medians?
5. What other methods can be used to prove angle relationships in geometric figures?

**Tip:** When working with triangle properties, remember that the centroid divides each median into a 2:1 ratio, which is crucial in many geometric proofs.

Cho hình vẽ sau, a) Chứng minh ΔAOB = ΔCOB, b) Chứng minh ∠AOB + ∠BOC = 180°.

Solve a geometric problem involving triangle congruence and angle sum properties. This problem explores key concepts like the Side-Angle-Side (SAS) criterion and the properties of medians in triangles. Suitable for students in Grades 8-10.

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