We are given two triangles that share two pairs of congruent angles, meaning they are similar triangles by the AA similarity criterion. From the diagram:

• Triangle $\triangle ABC$ is similar to $\triangle ABD$.
• We are tasked to find $AD$, given $AC = 10$ units.

Let’s break it down:

Step 1: Ratio of Similar Triangles

Since the triangles are similar, the sides are proportional: $\frac{AB}{AB} = \frac{AC}{AD}.$

Step 2: Relationship Between Lengths

Here, $AC = AB + BC$ and $AD = AB + BD$. Therefore, the proportion becomes: $\frac{AB}{AB} = \frac{AB + BC}{AB + BD}.$

StepApologies for the interruption earlier! Let us systematically solve the problem.

Step 1: Understand Triangle Similarity

We know the triangles $\triangle ABC$ and $\triangle ABD$ are similar because:

1. Angle $A$ is common.
2. Angle $B$ in $\triangle ABC$ is congruent to Angle $B$ in $\triangle ABD$ (marked in the diagram).

Thus, the triangles are similar by AA similarity.

Step 2: Proportionality of Sides

By similarity, corresponding side lengths are proportional: $\frac{AC}{AD} = \frac{AB}{AB}.$

From the diagram, $AC = AB + BC$ and $AD = AB + BD$.

Step 3: Known Information

It is given that $AC = 10$ units. Using proportional reasoning, we must calculate the side lengths to solve for $AD$.

Assuming the scale factor involves some geometric properties (e.g., if further dimensions or relationships were given), we’d find $AD$ proportionally using the same constant. Here, we can reason through possible options for congruence to solve directly or consider external hints.

Could you confirm or clarify any additional setup detail?

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