Exercise 37 Page 185

Let's begin by looking at the given information and the desired outcome of the proof. Then we can prove Theorem $3.4.$ We are given that $a\parallel b$ and $t\perp a.$ This is how we will begin our proof. We know that $t$ is perpendicular to $a.$ Therefore, by the definition of perpendicular lines, $\angle 1$ is a right angle. By the definition of a right angle, we know that the measure of $\angle 1$ is $90,$ $m\angle 1=90.$ Next, from the diagram we can tell that $\angle 1$ and $\angle 2$ are corresponding angles. Therefore, by the Corresponding Angles Theorem, $\angle 1\cong \angle 2.$ By the definition of congruent angles, we can tell that $m\angle 1=m\angle 2.$ Notice that now we can substitute $90$ for $m\angle 1$ in this equation. This gives us $90=m\angle 2,$ which is equivalent to $m\angle 2=90.$ The measure of $\angle 2$ is $90,$ so by the definition of a right angle, $\angle 2$ is a right angle. By the definition of perpendicular lines, we can conclude that the lines $t$ and $b$ are perpendicular, $t\perp b.$ This is what we wanted to prove! Finally, we can complete our proof.

Statements	Reasons
$a\parallel b,$ $t\perp a$	Given
$\angle 1$ is a right angle	Definition of perpendicular lines
$m\angle 1=90$	Definition of a right angle
$\angle 1\cong \angle 2$	Corresponding Angles Theorem
$m\angle 1=m\angle 2$	Definition of congruent angles
$m\angle 2=90$	Substitution Property of Equality
$\angle 2$ is a right angle	Definition of a right angle
$t\perp b$	Definition of perpendicular lines