From these principles, we can deduce that if ∠t is a right angle and m∠t > m∠r, then ∠r must be an acute angle (less than 90 degrees). Given the triangle's angle measures, we can use the information in statement 4 to conclude that the side opposite ∠t (which we can call pt) is shorter than the side opposite ∠r (which we can call pr), because ∠t is larger than ∠r. Hence, pt < pr.
The statements you've provided seem to describe a set of geometric principles and possibly steps in a geometric proof. Let's analyze the information step by step:
1. Perpendicular lines form right angles.
This principle tells us that if two lines are perpendicular, the angles where they intersect are right angles, which means they each measure 90 degrees.
2. ∠t is a right angle (given).
This is a given piece of information in your problem, establishing that ∠t is 90 degrees.
3. The measure of a right angle is greater than the measure of an acute angle.
By definition, an acute angle measures less than 90 degrees. Since a right angle is exactly 90 degrees, any acute angle is less than ∠t.
4. pt < pr if two angles of a triangle are not equal, then the side opposite the larger angle is longer than the side opposite the smaller angle.
This is a principle that relates the angles of a triangle to the lengths of its sides. It tells us that the larger the angle in a triangle, the longer the side opposite it.
From these principles, we can deduce that if ∠t is a right angle and m∠t > m∠r, then ∠r must be an acute angle (less than 90 degrees). Given the triangle's angle measures, we can use the information in statement 4 to conclude that the side opposite ∠t (which we can call pt) is shorter than the side opposite ∠r (which we can call pr), because ∠t is larger than ∠r. Hence, pt < pr.