Question

wy ⊥ xy. complete the proof that m∠w + m∠x = 90°.
(image of triangle xyw with a table of statements and reasons:

1. wy ⊥ xy (given)
2. m∠y = 90° (reason to complete)
3. m∠w + m∠x + m∠y = 180° (reason to complete)
4. m∠w + m∠x + 90° = 180° (substitution)
5. m∠w + m∠x = 90° (properties of addition, subtraction, multiplication, and division))

Explanation:

Step1: Analyze Statement 2

The reason for \( m\angle Y = 90^\circ \) is that if two lines are perpendicular, the angle between them is \( 90^\circ \). Since \( \overline{WY} \perp \overline{XY} \), \( \angle Y \) is the right angle formed by their intersection, so the reason is "Definition of perpendicular lines (if two lines are perpendicular, they form a right angle of \( 90^\circ \))".

Step2: Analyze Statement 3

The sum of the interior angles of a triangle is \( 180^\circ \). In triangle \( XYW \), the sum of \( m\angle W \), \( m\angle X \), and \( m\angle Y \) should be \( 180^\circ \), so the reason is "Triangle Angle Sum Theorem (the sum of the measures of the interior angles of a triangle is \( 180^\circ \))".

Answer:

For Statement 2, Reason: Definition of perpendicular lines (if two lines are perpendicular, they form a \( 90^\circ \) angle).
For Statement 3, Reason: Triangle Angle Sum Theorem (sum of interior angles of a triangle is \( 180^\circ \)).