Two or more <u>triangles</u> are <em>congruent </em>if on comparison, they have equal lengths of <u>sides,</u> and measure of <u>angles</u>.
Therefore, the required proofs for each question are shown below:
Problem 1:
<em>Congruent triangles</em> are <u>triangles</u> with equal lengths of <em>corresponding</em> <u>sides</u> and measures of internal <u>angles</u>.
Thus,
STATEMENT REASON
1. <NMQ ≅ <NPQ Any point on a <em>perpendicular bisector</em>
makes <u>equal</u> measure of angle with the
two ends of the<em> line</em> segment.
2. NQ ⊥ MP Definition of a<u> line</u>.
3. MQ ≅ PQ <em>Equal segments</em> of a bisected <u>line</u>.
4. MN ≅ PN Any point on a <em>perpendicular bisector </em>
is at the same <u>distance</u> to the
two ends of the <em>line segment</em>.
5. <MNQ ≅ <PNQ <u>Equal</u> measure of the <u>bisected</u> angle.
Problem 2:
A line <em>segment</em> is the shortest <u>distance</u> between two points.
STATEMENTS REASONS
1. m<PSR ≅ m<PSQ A <em>perpendicular bisector </em>is always at a right
angle to the <u>bisected</u> <em>line segment</em>.
2. m<RPS ≅ m<QPS Equal measure of the <u>bisected</u> <em>angle</em>.
3. RS ≅ QS Property of a <u>bisected</u> <em>line</em> segment.
4. PR ≅ PQ Any point on a <em>perpendicular bisector </em>
is at the same <u>distance</u> to the two ends of
the <u>line</u> segment.
For more clarifications on the perpendicular bisector of a line segment, visit: brainly.com/question/12475568
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