Answer:
Explanation:
Given:
QR || RS
TU intersects UV
m∠QRS = 90° (Definition of perpendicular lines)
m∠TUV = 90° (Definition of perpendicular lines)
m∠2QRS = m∠ZTUV
To prove: ZQRSZTUV
Proof:
QR || RS (Given)
TU intersects UV (Given)
m∠QRS = 90° (Definition of perpendicular lines)
m∠TUV = 90° (Definition of perpendicular lines)
m∠2QRS = m∠ZTUV (Given)
Now, let's continue with the proof:
Since m∠QRS = 90° and m∠TUV = 90° (from statements 3 and 4), we have a pair of complementary angles.
Complementary angles add up to 90°. Therefore, m∠QRS + m∠TUV = 90°.
By the Transitive Property of Equality, we can say that m∠2QRS + m∠TUV = 90° (from statements 5 and 7).
Now, we have established that m∠2QRS and m∠TUV together form a pair of complementary angles totaling 90°.
In a pair of complementary angles, if one angle is 90°, then the other angle must also be 90°.
Therefore, m∠2QRS = 90°.
Now, we have proved that m∠2QRS is a 90° angle, which means that ZQRS is a right angle.
Similarly, using the information from statement 4 (m∠TUV = 90°), we can conclude that ZTUV is a right angle.
Now, we have proved that both ZQRS and ZTUV are right angles, which means that the quadrilateral QRSZTUV has two pairs of opposite angles that are each 90°. Therefore, it is a rectangle.
So, we have completed the proof that ZQRSZTUV is a rectangle based on the given statements and reasons provided.