Question

QR 1 RS and TU I UV. Complete the proof that ZQRSZTUV.

1
Statement
1
QRL RS
1
(
2 TU I UV
3 m/QRS = 90°
4 m/TUV = 90°
5 m2QRS = MZTUV
6
mz
1
0
11
T
A
211
+
Reason
Given
Given
Definition of perpendicular lines
Definition of perpendicular lines
Transitive Property of Equality

Answer 1

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.