Question

problem 24
given: $overline{vu}paralleloverline{qr},overline{vt}paralleloverline{rs},$ $qt = us$
prove: $\triangle qrscong\triangle uvt$

1. $qt + st=us + st$
2. $qs = ut$
3. given
4. $angle vtucongangle qsr$

Explanation:

Step1: Justify step 5

Using addition property.

Step2: Justify step 6

Apply transitive property.

Step3: State given for step 7

List given parallel - lines.

Step4: Justify step 8

Use corresponding angles postulate.

Answer:

1. Reason: Addition Property of Equality (If \(a = b\), then \(a + c=b + c\). Here \(a = QT\), \(b = US\) and \(c = ST\)).
2. Statement: Since \(QT+ST = QS\) and \(US + ST=UT\) and \(QT + ST=US + ST\) (from step 5), by the Transitive Property of Equality, \(QS = UT\). Reason: Transitive Property of Equality.
3. Statement: \(\overline{VU}\parallel\overline{QR},\overline{VT}\parallel\overline{RS}\) (This is the given parallel - line information which will be used for angle - congruence).
4. Reason: Corresponding Angles Postulate (Because \(\overline{VT}\parallel\overline{RS}\), \(\angle VTU\) and \(\angle QSR\) are corresponding angles).