Question

$\overline{gh} \perp \overline{fh}$. complete the proof that $m\angle f + m\angle g = 90^\circ$.

(image of triangle fhg with h, f, g vertices)

	statement	reason
2	$m\angle h = 90^\circ$	definition of perpendicular lines
3	$m\angle f + m\angle g + m\angle h = 180^\circ$
4	$m\angle f + m\angle g + 90^\circ = 180^\circ$
5	$m\angle f + m\angle g = 90^\circ$	properties of addition, subtraction, multiplication, and division

Explanation:

Step1: Identify the triangle angle sum property

The sum of the interior angles of a triangle is always \(180^\circ\). In triangle \(FGH\), the angles are \(\angle F\), \(\angle G\), and \(\angle H\), so \(m\angle F + m\angle G + m\angle H = 180^\circ\) by the Triangle Angle Sum Theorem.

Step2: Substitute the known angle measure

We know from step 2 that \(m\angle H = 90^\circ\). So we substitute \(m\angle H\) with \(90^\circ\) in the equation from step 3. This is the Substitution Property (since we replace \(m\angle H\) with its known value \(90^\circ\)).

For row 3, the reason is "Triangle Angle Sum Theorem" (the sum of interior angles of a triangle is \(180^\circ\)). For row 4, the reason is "Substitution Property" (substituting \(m\angle H = 90^\circ\) into the equation from row 3).

Answer:

• Row 3 Reason: Triangle Angle Sum Theorem
• Row 4 Reason: Substitution Property