Question

part b
if $angle r = 110^circ$, $angle z = 35^circ$ and $angle pst = 40^circ$, is $\triangle wsr sim \triangle zsp$?
explain your reasoning.
④ if one angle of $\triangle wsr$ is congruent to one angle of $\triangle zsp$

Explanation:

Step1: Find ∠W in △WSR

The sum of angles in a triangle is $180^\circ$.
$\angle W = 180^\circ - \angle R - \angle WSR$
First, note $\angle WSR = 180^\circ - \angle PST = 180^\circ - 40^\circ = 140^\circ$
$\angle W = 180^\circ - 110^\circ - 140^\circ$ → This is invalid, so correct: $\angle WSR$ is supplementary to $\angle PST$? No, correction: $\angle WSR$ and $\angle PST$ are not supplementary. Wait, $\angle WSR$ is an angle of $\triangle WSR$, so $\angle W = 180^\circ - 110^\circ - \angle WSR$. But $\angle WSR$ corresponds to $\angle ZSP$? No, first find all angles of $\triangle ZSP$:

Step1: Calculate ∠SPZ in △ZSP

Sum of angles in triangle is $180^\circ$.
$\angle SPZ = 180^\circ - \angle Z - \angle PST$
$\angle SPZ = 180^\circ - 35^\circ - 40^\circ = 105^\circ$

Step2: Calculate ∠W in △WSR

Sum of angles in triangle is $180^\circ$.
$\angle W = 180^\circ - \angle R - \angle WSR$
Assume $\angle WSR = \angle PST = 40^\circ$ (vertical angles)
$\angle W = 180^\circ - 110^\circ - 40^\circ = 30^\circ$

Step3: Compare corresponding angles

For similarity, all pairs of corresponding angles must be equal.
$\triangle WSR$ angles: $110^\circ, 40^\circ, 30^\circ$
$\triangle ZSP$ angles: $35^\circ, 40^\circ, 105^\circ$
No three matching equal angle pairs.

Answer:

No, $\triangle WSR$ is not similar to $\triangle ZSP$. The angles of $\triangle WSR$ are $110^\circ$, $40^\circ$, and $30^\circ$, while the angles of $\triangle ZSP$ are $35^\circ$, $40^\circ$, and $105^\circ$. Since the three pairs of corresponding angles are not all congruent, the triangles do not satisfy the AA (Angle-Angle) similarity criterion.