Question

In the two triangles, what is the value of \( P + Q + R + S \)?

I. \( A + B = 90^\circ \) 
II. \( P + Q = R + S \)

• A. If the statement I alone is sufficient to answer the question.
• B. If the statement II alone is sufficient to answer the question.
• C. If the statements I and II together are sufficient to answer the question but neither statement alone is sufficient.
• D. If the statements I and II together are not sufficient to answer the question and additional data is required.

Hint:

In geometric problems involving triangles, remember that the sum of the angles in any triangle is always 180 degrees.

Answer 1

The Correct Option is A

We are given two triangles with angles \( P, Q, R, S \), and other conditions. - From condition I: \( A + B = 90^\circ \), this means the angles \( A \) and \( B \) in the first triangle add up to 90 degrees. This implies that the remaining two angles in the first triangle must also add up to 90 degrees. - From condition II: \( P + Q = R + S \), this equation means that the sum of angles \( P \) and \( Q \) is equal to the sum of angles \( R \) and \( S \). In any triangle, the sum of the interior angles is always \( 180^\circ \). Therefore, in each triangle, the sum of the angles must be \( 180^\circ \). Thus: \[ P + Q + R + S = 180^\circ + 180^\circ = 360^\circ \] Therefore, the value of \( P + Q + R + S \) is \( 360^\circ \). Thus, the correct answer is \( \boxed{1} \).