Question

complete the proof that $\angle ptq \cong \angle rts$.
image of intersecting lines with points r, s, q, p, t

	statement	reason
1	$m\angle ptq + m\angle qtr = 180^\circ$
2	$m\angle qtr + m\angle rts = 180^\circ$
3	$m\angle ptq + m\angle qtr = m\angle qtr + m\angle rts$
4	$m\angle ptq = m\angle rts$
5	$\angle ptq \cong \angle rts$

Explanation:

Step1: Identify the reason for Statement 1

Angles that form a linear pair are supplementary (sum to \(180^\circ\)). \(\angle PTQ\) and \(\angle QTR\) form a linear pair (they are adjacent and their non - common sides form a straight line), so the reason for \(m\angle PTQ + m\angle QTR=180^\circ\) is "Linear Pair Postulate" (angles forming a linear pair are supplementary).

Step2: Reason for Statement 2

Similarly, \(\angle QTR\) and \(\angle RTS\) form a linear pair. So the reason for \(m\angle QTR + m\angle RTS = 180^\circ\) is also "Linear Pair Postulate".

Step3: Reason for Statement 3

We know that \(m\angle PTQ + m\angle QTR = 180^\circ\) (from Statement 1) and \(m\angle QTR + m\angle RTS=180^\circ\) (from Statement 2). By the "Transitive Property of Equality" (if \(a = c\) and \(b = c\), then \(a = b\); here \(a=m\angle PTQ + m\angle QTR\), \(b = m\angle QTR + m\angle RTS\), and \(c = 180^\circ\)), we can say \(m\angle PTQ + m\angle QTR=m\angle QTR + m\angle RTS\).

Step4: Reason for Statement 4

To get \(m\angle PTQ=m\angle RTS\) from \(m\angle PTQ + m\angle QTR=m\angle QTR + m\angle RTS\), we use the "Subtraction Property of Equality" (subtract \(m\angle QTR\) from both sides of the equation).

Step5: Reason for Statement 5

If two angles have equal measures, then they are congruent. So the reason for \(\angle PTQ\cong\angle RTS\) is "Definition of Congruent Angles" (angles with equal measures are congruent).

Answer:

1. Reason for Statement 1: Linear Pair Postulate (angles forming a linear pair are supplementary)
2. Reason for Statement 2: Linear Pair Postulate (angles forming a linear pair are supplementary)
3. Reason for Statement 3: Transitive Property of Equality
4. Reason for Statement 4: Subtraction Property of Equality
5. Reason for Statement 5: Definition of Congruent Angles (angles with equal measures are congruent)