Question

$\overline{yz} \perp \overline{xz}$. complete the proof that $m\angle x + m\angle y = 90^\circ$.
(image of triangle xyz with z as the right angle vertex)

	statement	reason
1	$\overline{yz} \perp \overline{xz}$
2	$m\angle z = 90^\circ$
3	$m\angle x + m\angle y + m\angle z = 180^\circ$
4	$m\angle x + m\angle y + 90^\circ = 180^\circ$
5	$m\angle x + m\angle y = 90^\circ$

Explanation:

Step1: Identify Given

Given \(\overline{YZ} \perp \overline{XZ}\), this is the starting info.

Step2: Define Perpendicular Angles

If two lines are perpendicular, they form a \(90^\circ\) angle. So \(m\angle Z = 90^\circ\) (Definition of perpendicular lines).

Step3: Triangle Angle Sum

The sum of angles in a triangle is \(180^\circ\) (Triangle Angle Sum Theorem), so \(m\angle X + m\angle Y + m\angle Z = 180^\circ\).

Step4: Substitute \(m\angle Z\)

Substitute \(m\angle Z = 90^\circ\) into the angle - sum equation (Substitution Property of Equality), getting \(m\angle X + m\angle Y + 90^\circ = 180^\circ\).

Step5: Solve for \(m\angle X + m\angle Y\)

Subtract \(90^\circ\) from both sides of the equation (Subtraction Property of Equality): \(m\angle X + m\angle Y=180^\circ - 90^\circ = 90^\circ\).

Answer:

The reasons for each step are:

1. Given
2. Definition of perpendicular lines (if two lines are perpendicular, the angle between them is \(90^\circ\))
3. Triangle Angle Sum Theorem (the sum of the interior angles of a triangle is \(180^\circ\))
4. Substitution Property of Equality (substitute \(m\angle Z = 90^\circ\) into \(m\angle X + m\angle Y + m\angle Z = 180^\circ\))
5. Subtraction Property of Equality (subtract \(90^\circ\) from both sides of \(m\angle X + m\angle Y + 90^\circ = 180^\circ\))

And we have proved that \(m\angle X + m\angle Y = 90^\circ\)