roblem 11 - Attempted Problem: A straight line $$ \ell $$ divides a triangle $$ \Delta $$ into two congruent triangles. Select all the statements that must be true. Triangle $$ \Delta $$ has two equal sides. Triangle $$ \Delta $$ has two equal angles. The line $$ \ell $$ is perpendicular to a side of $$ \Delta $$. The line $$ \ell $$ passes through a midpoint of one side of $$ \Delta $$. .

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To analyze the problem, we need to understand the implications of a straight line $$ \ell $$ dividing a triangle $$ \Delta $$ into two congruent triangles.

### Known Conditions:
1. The line $$ \ell $$ divides triangle $$ \Delta $$ into two congruent triangles.
2. Congruent triangles have equal corresponding sides and angles.

### Analyzing Each Statement:

1. **Triangle $$ \Delta $$ has two equal sides.**
   - This statement is **not necessarily true**. A triangle can be scalene (no equal sides) and still be divided into two congruent triangles by a line. For example, if the line divides the triangle into two equal areas without requiring equal sides, this statement does not hold.

2. **Triangle $$ \Delta $$ has two equal angles.**
   - This statement is also **not necessarily true**. Similar to the first statement, a triangle can have different angles and still be divided into two congruent triangles. The congruence of the resulting triangles does not imply that the original triangle has equal angles.

3. **The line $$ \ell $$ is perpendicular to a side of $$ \Delta $$.**
   - This statement is **not necessarily true**. The line can divide the triangle into two congruent triangles without being perpendicular to any side. For instance, a line can bisect the area of a triangle without being perpendicular.

4. **The line $$ \ell $$ passes through a midpoint of one side of $$ \Delta $$.**
   - This statement is **true**. For a line to divide a triangle into two congruent triangles, it must pass through the midpoint of one of the sides. This ensures that the areas of the two resulting triangles are equal.

### Conclusion:
The only statement that must be true is:

- **The line $$ \ell $$ passes through a midpoint of one side of $$ \Delta $$.**

The other statements do not necessarily hold true in all cases.
The line $$ \ell $$ must pass through the midpoint of one side of triangle $$ \Delta $$.

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