Respuesta :
We are given that the triangles ABC and QPR are similar.
Recall the properties of similar triangles,
• Corresponding angles are equal.
,• Corresponding sides are in the same ratio.
Let us first identify the corresponding sides
Side AB = Side QP
Side BC = Side RP
Side AC = Side QR
Now let us find the missing side length RP using the ratio of corresponding sides.
[tex]\begin{gathered} \frac{QP}{AB}=\frac{RP}{BC} \\ \frac{5}{10}=\frac{RP}{9} \\ RP=\frac{5}{10}\cdot9 \\ RP=4.5 \end{gathered}[/tex]So, the length of the side RP is 4.5 hence the 2nd option is correct.
Now let us check the corresponding angles.
m∠R = m∠C
m∠Q = m∠A
m∠P = m∠B
From the figure, we see that
[tex]\begin{gathered} m\angle R=m\angle C=81\degree \\ m\angle Q=m\angle A=63\degree \end{gathered}[/tex]Hence the 1st option and the 3rd option are correct.
Recall that the sum of the interior angles of a triangle is equal to 180°
[tex]\begin{gathered} m\angle R+m\angle Q+m\angle P=180\degree \\ 81\degree+63\degree+m\angle P=180\degree \\ 144\degree+m\angle P=180\degree \\ m\angle P=180\degree-144\degree \\ m\angle P=36\degree \end{gathered}[/tex]Hence, the last option is incorrect since the m∠P is 36° (not 81°)
Therefore, the correct options are 1st, 2nd, and 3rd only.
