Question

since ad = be and db = ec, the proportion $\frac{ad}{db}$ is the same as $\frac{be}{ec}$, so the ratio of the rise to the run is the same in each triangle.
△adb is similar to △bec because the corresponding angles are congruent. therefore, $\frac{ad}{db}=\frac{be}{ec}$, so the ratio of the rise to the run is the same in each triangle.
△adb is similar to △bec because the corresponding angles are congruent. therefore, ab = bc, so the slope of $overline{ab}$ is equal to the slope of $overline{bc}$.
△adb is congruent to △bec because the corresponding angles are congruent, so the ratio of the rise to the run is the same in each triangle.

Explanation:

Step1: Recall similarity - angle criterion

If two triangles have corresponding - congruent angles, they are similar. In right - angled triangles formed on a coordinate plane, the ratio of rise to run is related to the slope.

Step2: Analyze the given information

We know that in right - triangles $\triangle ADB$ and $\triangle BEC$, the corresponding angles are congruent (since we are dealing with right - angled triangles formed by the grid and the line, and the angles with the x - axis are equal). For similar triangles, the ratios of corresponding sides are equal. The ratio of the rise to the run of a line segment is given by $\frac{\text{rise}}{\text{run}}$. In $\triangle ADB$, the rise is $AD$ and the run is $DB$, and in $\triangle BEC$, the rise is $BE$ and the run is $EC$. Since $\triangle ADB\sim\triangle BEC$, we have $\frac{AD}{DB}=\frac{BE}{EC}$.

Step3: Evaluate each statement

• The first statement: Just stating $AD = BE$ and $DB = EC$ is not the correct reason for the equal ratio of rise to run. The key is similarity due to congruent angles.
• The second statement: $\triangle ADB$ is similar to $\triangle BEC$ because the corresponding angles are congruent. And for similar triangles, $\frac{AD}{DB}=\frac{BE}{EC}$, which means the ratio of the rise to the run is the same in each triangle. This is correct.
• The third statement: Just because $\triangle ADB\sim\triangle BEC$ does not mean $AB = BC$. The correct relationship is about the ratio of the sides related to the rise - run concept.
• The fourth statement: The triangles are similar, not congruent. Congruence requires more than just congruent angles (e.g., side - side - side, side - angle - side, etc. criteria).

Answer:

The second statement " $\triangle ADB$ is similar to $\triangle BEC$ because the corresponding angles are congruent. Therefore, $\frac{AD}{DB}=\frac{BE}{EC}$, so the ratio of the rise to the run is the same in each triangle." is correct.