Let's go through the geometry problem step by step based on the image.

a. Why are points D, A, and E collinear?

Points $D$, $A$, and $E$ are collinear because of the congruence of the triangles and the shared line segments. Here's why:

• Since triangle $ABC$ is congruent to triangles $BAD$ and $CEA$, the line $DA$ in triangle $BAD$ corresponds to line $AE$ in triangle $CEA$.
• Both $DA$ and $AE$ are part of the same straight line that passes through point $A$. Thus, points $D$, $A$, and $E$ must lie on the same line, making them collinear.

b. Why is line $DE$ parallel to line $BC$?

Line $DE$ is parallel to line $BC$ due to the congruence between the triangles. Here’s the explanation:

• Since triangle $ABC$ is congruent to triangles $BAD$ and $CEA$, the angles corresponding to line segments $BC$ and $DE$ must be congruent.
• In both triangles $BAD$ and $CEA$, the corresponding angles at points $D$ and $E$ must match the angles at points $B$ and $C$, respectively.
• By the Corresponding Angles Postulate, if the corresponding angles are equal, the lines $DE$ and $BC$ must be parallel.

Would you like more details or further explanation on any part of this solution?

Here are 5 related questions to deepen understanding:

1. What are the properties of congruent triangles?
2. How does the Corresponding Angles Postulate help in proving lines are parallel?
3. Can you prove the collinearity of points using vector geometry?
4. How can the transversals between parallel lines help identify corresponding angles?
5. What are the conditions necessary for two lines to be parallel in Euclidean geometry?

Tip: In geometry, when working with congruent triangles, always look for corresponding angles and sides to infer additional properties like parallelism and collinearity.