Let's break down the math problems visible in the image and solve them step-by-step.

Problem 6:

In this diagram, we are given that:

• $AB \perp AC$ and $DE \perp BC$

(a) Why must $\triangle BAC$ be similar to $\triangle DEC$?

We can use the Angle-Angle (AA) Similarity Criterion to prove similarity between triangles.

• Both triangles share angle $\angle A$ (common angle).
• Since both $AB$ and $DE$ are perpendicular to $BC$, the angles $\angle ABC$ and $\angle DEC$ are right angles. Therefore, $\triangle BAC$ and $\triangle DEC$ have two angles equal (right angles and the shared angle $\angle A$).

By the AA similarity criterion, $\triangle BAC \sim \triangle DEC$.

(b) If $BC = 20$, $DE = 4$, and $DC = 8$, find the length of $AB$.

Using the similarity of the triangles $BAC$ and $DEC$, the sides of the triangles are proportional. This gives us the equation:

$\frac{AB}{DE} = \frac{BC}{DC}$

Substitute the known values:

$\frac{AB}{4} = \frac{20}{8}$

Simplifying the ratio:

$\frac{AB}{4} = 2.5$

Now, solve for $AB$:

$AB = 4 \times 2.5 = 10$

Thus, $AB = 10$.

Would you like more details on any of these steps, or have any questions? Here are some related questions to explore further:

1. Can you apply the Side-Side-Side (SSS) similarity criterion to these triangles?
2. How would you calculate other sides in $\triangle BAC$ using this similarity?
3. What is the geometric meaning of perpendicularity in triangle similarity?
4. How can we prove triangle similarity using more than two criteria?
5. Can you identify any other pairs of similar triangles in the diagram?

Tip: When using the similarity criterion, ensure that corresponding angles and sides are correctly identified to avoid mistakes in proportions.