Question

1. complete each statement that is justified by the given reason.

a. if \\( \overline { p q } \cong \overline { r s } \\), then ____. (definition of congruent segments)
b. if \\( \angle m \\) and \\( \angle n \\) form a linear pair, they are ____. (linear pair theorem)
c. if point \\( m \\) is the midpoint of \\( \overline { x y } \\), then ____. (midpoint theorem)
d. if \\( t u + u v = t v \\), then ____. (definition of between)

Explanation:

Part a

Step1: Recall the definition of congruent segments

Congruent segments have equal lengths. If \(\overline{PQ}\cong\overline{RS}\), by the definition of congruent segments, the lengths of the segments are equal.
\(PQ = RS\)

Part b

Step1: Recall the Linear Pair Theorem

The Linear Pair Theorem states that if two angles form a linear pair, then they are supplementary. Supplementary angles are two angles whose sum is \(180^{\circ}\).

Part c

Step1: Recall the Midpoint Theorem

The Midpoint Theorem (or the definition of a mid - point of a segment) states that if \(M\) is the midpoint of \(\overline{XY}\), then \(XM=MY\) (or \(XM=\frac{1}{2}XY\) and \(MY = \frac{1}{2}XY\))

Part d

Step1: Recall the definition of between

If \(TU+UV = TV\), by the definition of between (for points on a line segment), point \(U\) is between \(T\) and \(V\)

Answer:

a. \(PQ = RS\)
b. supplementary
c. \(XM = MY\) (or \(XM=\frac{1}{2}XY\) and \(MY=\frac{1}{2}XY\))
d. \(U\) is between \(T\) and \(V\)