Question

complete the proof that $m\angle s + m\angle t = 90^\circ$.

(image of triangle $stu$ with right angle at $u$)

	statement	reason
2	$m\angle u = 90^\circ$
3	$m\angle s + m\angle t + m\angle u = 180^\circ$
4	$m\angle s + m\angle t + 90^\circ = 180^\circ$
5	$m\angle s + m\angle t = 90^\circ$

Explanation:

Step1: Reason for \( m\angle U = 90^\circ \)

If two lines are perpendicular, the angle between them is \( 90^\circ \). Since \( \overline{TU} \perp \overline{SU} \), by the definition of perpendicular lines, \( m\angle U = 90^\circ \).

Step2: Reason for \( m\angle S + m\angle T + m\angle U = 180^\circ \)

The sum of the interior angles of a triangle is \( 180^\circ \). Triangle \( STU \) has angles \( \angle S \), \( \angle T \), and \( \angle U \), so by the Triangle Angle - Sum Theorem, \( m\angle S + m\angle T + m\angle U = 180^\circ \).

Step3: Reason for \( m\angle S + m\angle T + 90^\circ = 180^\circ \)

We know from Step 2 that \( m\angle S + m\angle T + m\angle U = 180^\circ \) and from Step 1 that \( m\angle U = 90^\circ \). By the Substitution Property of Equality, we substitute \( m\angle U \) with \( 90^\circ \) in the equation \( m\angle S + m\angle T + m\angle U = 180^\circ \) to get \( m\angle S + m\angle T + 90^\circ = 180^\circ \).

Step4: Reason for \( m\angle S + m\angle T = 90^\circ \)

Starting with the equation \( m\angle S + m\angle T + 90^\circ = 180^\circ \), we can use the Subtraction Property of Equality. Subtract \( 90^\circ \) from both sides of the equation: \( (m\angle S + m\angle T + 90^\circ)- 90^\circ=180^\circ - 90^\circ \), which simplifies to \( m\angle S + m\angle T = 90^\circ \).

Answer:

• For row 2: Reason is "Definition of perpendicular lines"
• For row 3: Reason is "Triangle Angle - Sum Theorem"
• For row 4: Reason is "Substitution Property of Equality"
• For row 5: Reason is "Subtraction Property of Equality"