similar triangles with segments

1. Identify the geometric properties: We have two right-angled triangles, $\triangle ABC$ and $\triangle DEC$. The angles $\angle ACB$ and $\angle DCE$ are marked as equal. Let $\angle ACB = \angle DCE = \theta$. Since $\angle BAC = 90^\circ$ and $\angle DEC = 90^\circ$, the triangles are similar by the Angle-Angle (AA) similarity criterion.

2. Set up the ratio of corresponding sides: Because $\triangle ABC \sim \triangle DEC$, the ratios of their corresponding legs must be equal. Specifically, the ratio of the vertical leg to the horizontal leg in both triangles is equal to $\tan(\theta)$:
$$\frac{AB}{AC} = \frac{DE}{CE}$$

3. Substitute the given values: We are given $AB = 84$, $AC = 156 - x$, $DE = 7$, and $CE = x$. Substituting these into the ratio equation:
$$\frac{84}{156 - x} = \frac{7}{x}$$

4. Solve for x: Cross-multiply to solve the equation:
$$84x = 7(156 - x)$$
$$84x = 1092 - 7x$$
$$84x + 7x = 1092$$
$$91x = 1092$$
$$x = \frac{1092}{91} = 12$$

5. Calculate the length of AC: The problem asks for the length of $\overline{AC}$. We know $AC = 156 - x$. Substituting $x = 12$:
$$AC = 156 - 12 = 144$$