The problem states:

In quadrilateral $PQRS$:

• $PQ = PS$ (indicating that $PQRS$ could potentially be an isosceles trapezoid or some other special quadrilateral),
• $PQ \parallel RS$,
• $\angle Q = 60^\circ$,
• $\angle S = 90^\circ$.

We are to find the other two angles, $\angle P$ and $\angle R$.

Solution

1. Since $PQ \parallel RS$, we know that $PQRS$ is a trapezoid.

2. In any quadrilateral, the sum of the interior angles is $360^\circ$.

3. Therefore, we have: $\angle Q + \angle S + \angle P + \angle R = 360^\circ$

4. Substituting the known values: $60^\circ + 90^\circ + \angle P + \angle R = 360^\circ$ Simplifying: $150^\circ + \angle P + \angle R = 360^\circ$ $\angle P + \angle R = 210^\circ$

5. In an isosceles trapezoid with $PQ = PS$ and $PQ \parallel RS$, opposite angles (formed by non-parallel sides) are supplementary. Thus: $\angle Q + \angle R = 180^\circ \quad \text{and} \quad \angle S + \angle P = 180^\circ$

6. Since $\angle Q = 60^\circ$, we have: $\angle R = 180^\circ - 60^\circ = 120^\circ$

7. Since $\angle S = 90^\circ$, we have: $\angle P = 180^\circ - 90^\circ = 90^\circ$

Final Answer

The other two angles are:

• $\angle P = 90^\circ$
• $\angle R = 120^\circ$

Would you like further details or have any questions?

Here are five related questions to expand on this topic:

1. How do you determine the sum of angles in any quadrilateral?
2. What are the properties of an isosceles trapezoid?
3. Why are opposite angles supplementary in an isosceles trapezoid?
4. How can parallel sides in a quadrilateral help in angle calculations?
5. What differentiates an isosceles trapezoid from other quadrilaterals?

Tip: In any quadrilateral with parallel sides, using properties of supplementary angles can simplify angle calculations significantly.