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 \( \Delta \). .

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.