Show that triangle ADF is similar to triangle ABE and complete the table for y = 2x − 3.

14. To show that $\triangle \text{ADF}$ and $\triangle \text{ABE}$ are similar:

Step 1: Identify properties of the square ABCD.
Since ABCD is a square, all its angles are $90^\circ$.
Therefore, $\angle \text{ADF} = 90^\circ$ and $\angle \text{ABE} = 90^\circ$.
So, $\angle \text{ADF} = \angle \text{ABE}$.

Step 2: Use the given condition.
It is given that $\angle \text{AEB} = \angle \text{AFD}$.

Step 3: Apply the Angle-Angle (AA) similarity criterion.
In $\triangle \text{ADF}$ and $\triangle \text{ABE}$:

• $\angle \text{ADF} = \angle \text{ABE}$ (Both are $90^\circ$ angles of the square).
• $\angle \text{AFD} = \angle \text{AEB}$ (Given).

Since two pairs of corresponding angles are equal, the third pair of angles must also be equal ($\angle \text{DAF} = \angle \text{BAE}$).
Therefore, by the AA similarity criterion, $\triangle \text{ADF}$ is similar to $\triangle \text{ABE}$.

15. The table below shows the values of $x$ and $y$ for the equation $y = 2x - 3$.