Question

select the correct answer from each drop - down menu.
a student is given $\angle qos\cong\angle rot$ and the diagram and asked to prove $m\angle qor = 56^\circ$
given: $\angle qos\cong\angle rot$
prove: $m\angle qor = 56^\circ$
here is the student’s proof.

1. $\angle qos\cong\angle tor$ given
2. $m\angle qos = m\angle rot$ definition of congruence
3. $m\angle qor + m\angle ros=m\angle tos + m\angle ros$ angle addition
4. $m\angle qor = m\angle tos$ subtraction property of congruence
5. $m\angle tos = 56^\circ$ given
6. $m\angle qor = 56^\circ$ subtraction property of equality

where is the first error or skipped step in the proof?
the reason for statement is not correct
because

Explanation:

To find the first error or skipped step, we analyze each step:

1. Step 1: Given \(\angle QOS \cong \angle ROT\) (valid, as it’s given).
2. Step 2: \(m\angle QOS = m\angle ROT\) (valid, by definition of congruent angles).
3. Step 3: \(m\angle QOR + m\angle ROS = m\angle TOS + m\angle ROS\) – Wait, the diagram shows \(\angle QOS\) and \(\angle ROT\) share \(\angle ROS\), but the angle - addition here is misapplied. However, the key error is in Step 4: “\(m\angle QOR = m\angle TOS\) (subtraction property of congruence)”. The subtraction property of equality (not congruence) applies here, since we subtract \(m\angle ROS\) from both sides of \(m\angle QOR + m\angle ROS = m\angle TOS + m\angle ROS\). Also, Step 5: “\(m\angle TOS = 56^\circ\) (given)” – but \(m\angle TOS\) is not given; \(m\angle QOS = 56^\circ\) (from the diagram, \(\angle QOS\) is labeled \(56^\circ\)). So the first error is Step 4: incorrect reason (uses “subtraction property of congruence” instead of “subtraction property of equality”) and Step 5: incorrect “given” (since \(m\angle TOS\) is not given, \(m\angle QOS = 56^\circ\) is given, and we should relate \(m\angle QOR\) to \(m\angle QOS\) via congruence). But the first error in reasoning is Step 4: the reason “subtraction property of congruence” is wrong; it should be subtraction property of equality. Also, Step 5 misstates what is given. However, the first error in the proof (logical error) is Step 4: the reason for \(m\angle QOR = m\angle TOS\) is incorrect (congruence is for angles, subtraction of measures uses equality properties).

Answer:

The first error is in Step 4: The reason “subtraction property of congruence” is incorrect. It should be the subtraction property of equality (since we are subtracting angle measures, which are real numbers, so equality properties apply, not congruence properties). Additionally, Step 5 incorrectly claims \(m\angle TOS = 56^\circ\) is “given”; \(m\angle QOS = 56^\circ\) is given, and \(m\angle TOS\) is equal to \(m\angle QOR\) (from Step 4, if corrected) and \(m\angle QOS = 56^\circ\) (from \(\angle QOS\)’s measure in the diagram). The first error in the proof (the first step with a wrong reason or statement) is Step 4, where the subtraction property of congruence is misused (should be subtraction property of equality).