Question

State the principal theorem that justifies the statement about the diagram.

If D is the midpoint of $\overline{\mathrm{AB}}$, then $\overline{\mathrm{DE}}\parallel \overline{\mathrm{BC}}$.

Short answer

The theorem that justifies the statement about the diagram istheorem 5-11 which states that the segment that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

It is being given that D is the midpoint of $\overline{AB}$.

From the given diagram it can be observed that $AE=9$and $EC=9$.

Therefore, $AE=EC=9$.

Therefore, E is the midpoint of AC.

Theorem 5-11 states that the segment that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side.

As it can be seen that DE is a segment that joins the midpoint D of the side AB and the midpoint E of the side AC therefore by using the theorem 5-11 it can be said that as DE is a segment that joins the midpoints of two sides of a triangle therefore DE will be parallel to the third side BC.

Therefore, $DE\parallel BC$.

Step 3. Write the theorem that justifies the statement about the diagram.

The theorem that justifies the statement about the diagram is theorem 5-11 which states that the segment that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side.