If both \(ABDC\) and \(CPDFE\) are parallelograms, what is \(q + r\)? (1) r = 70◦ (2) p = 110◦
Show Hint
In parallelogram geometry, opposite angles are equal and adjacent angles are supplementary — these two rules can directly solve many angle-sum problems.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient.
EACH statement ALONE is sufficient.
Hide Solution
Verified By Collegedunia
The Correct Option isD
Solution and Explanation
From Statement (1):
Given \(r = 70^\circ\).
In parallelogram \(ABDC\), angles \(q\) and \(r\) are adjacent:
\[
q = 180^\circ - r = 180^\circ - 70^\circ = 110^\circ
\]
Thus:
\[
q + r = 110^\circ + 70^\circ = 180^\circ
\]
We found \(q + r\) from Statement (1) alone, so it is sufficient.
From Statement (2):
Given \(p = 110^\circ\).
In parallelogram \(CPDFE\), \(p\) and \(q\) are adjacent:
\[
q = 180^\circ - p = 180^\circ - 110^\circ = 70^\circ
\]
In parallelogram \(ABDC\), \(q\) and \(r\) are opposite angles:
\[
r = q = 70^\circ
\]
Thus:
\[
q + r = 70^\circ + 70^\circ = 140^\circ
\]
We found \(q + r\) from Statement (2) alone, so it is also sufficient.
Since each statement alone is sufficient, the answer is \(\boxed{\text{D}}\).
