Question

determine if triangle nop and triangle qrs are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

Explanation:

Step1: Check the ratio of corresponding sides

Find the ratios of the corresponding sides. For side $NP = 26$ and $QS=52$, the ratio is $\frac{QS}{NP}=\frac{52}{26} = 2$. For side $NO = 20$ and $QR = 40$, the ratio is $\frac{QR}{NO}=\frac{40}{20}=2$.

Step2: Check the included - angle

The included angle $\angle N=49^{\circ}$ and $\angle Q = 49^{\circ}$, so the included angles are equal.

Step3: Apply the similarity criterion

Since the ratios of two pairs of corresponding sides are equal ($\frac{QS}{NP}=\frac{QR}{NO} = 2$) and the included angles are equal ($\angle N=\angle Q$), by the Side - Angle - Side (SAS) similarity criterion, the two triangles are similar.

Answer:

The triangles $NOP$ and $QRS$ are similar by the Side - Angle - Side (SAS) similarity criterion.