Question

supplement theorem
given: \\( \angle pqt \\) and \\( \angle tqr \\) form a linear pair.
prove: \\( \angle pqt \\) and \\( \angle tqr \\) are supplementary.

proof:

statements	reasons
2. ?	2. given from figure
3. ?	3. def. of straight angle
4. \\( m\angle pqt + m\angle tqr = m\angle pqr \\)	4. ?
5. ?	5. substitution
6. \\( \angle pqt \\) and \\( \angle tqr \\) are supplementary	6. ?

keyboard help

Explanation:

Step1: Reason for statement 1

The first statement is given in the problem, so the reason is "Given".

Step2: Statement 2

From the figure, we can observe that \(\angle PQR\) is a straight angle. So the statement is "\(\angle PQR\) is a straight angle".

Step3: Statement 3

By the definition of a straight angle, \(m\angle PQR = 180^{\circ}\). So the statement is "\(m\angle PQR=180^{\circ}\)".

Step4: Reason for statement 4

This is based on the Angle - Addition Postulate.

Step5: Statement 5

Substituting \(m\angle PQR = 180^{\circ}\) into \(m\angle PQT + m\angle TQR=m\angle PQR\), we get \(m\angle PQT + m\angle TQR = 180^{\circ}\).

Step6: Reason for statement 6

By the definition of supplementary angles (if the sum of the measures of two angles is \(180^{\circ}\), then the angles are supplementary).

Answer:

1. Given
2. \(\angle PQR\) is a straight angle
3. \(m\angle PQR = 180^{\circ}\)
4. Angle - Addition Postulate
5. \(m\angle PQT + m\angle TQR=180^{\circ}\)
6. Definition of supplementary angles