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