Question

uw ⊥ vw. complete the proof that m∠u + m∠v = 90°.

(image of triangle uvw)

statement	reason
2. ( mangle w = 90^circ )
3. ( mangle u + mangle v + mangle w = 180^circ )	additive property of angle measure
4. ( mangle u + mangle v + 90^circ = 180^circ )
5. ( mangle u + mangle v = 90^circ )

(reasons to choose from: additive property of angle measure, additive property of length, all right angles are congruent, angles forming a linear pair sum to 180°, definition of angle bisector, cut off text)

Explanation:

Step1: Analyze Statement 2

We know that if two lines are perpendicular (given \(\overline{UW} \perp \overline{VW}\)), then the angle between them is a right angle. By the definition of perpendicular lines, a right angle has a measure of \(90^\circ\). So the reason for \(m\angle W = 90^\circ\) is the definition of perpendicular lines (which implies a right angle, and a right angle measures \(90^\circ\)). But looking at the dropdown for the reason of statement 2, wait, actually in the table, statement 2 is \(m\angle W = 90^\circ\) and we need to find its reason. Since \(\overline{UW} \perp \overline{VW}\), the definition of perpendicular lines tells us that the angle between them (\(\angle W\)) is \(90^\circ\), which is also the definition of a right angle (a right angle is \(90^\circ\)). But from the options, the relevant one for why \(m\angle W = 90^\circ\) when \(\overline{UW} \perp \overline{VW}\) is the definition of perpendicular lines leading to a right angle, but among the given options, the closest is "Definition of perpendicular lines (implies right angle, so \(m\angle W = 90^\circ\))", but in the dropdown, the options include "All right angles are congruent" no, wait, the correct reason for \(m\angle W = 90^\circ\) given \(\overline{UW} \perp \overline{VW}\) is the definition of perpendicular lines (which states that if two lines are perpendicular, they form a right angle, and a right angle has measure \(90^\circ\)). But in the provided options, maybe the intended reason is that perpendicular lines form a right angle, so \(m\angle W = 90^\circ\) by definition of perpendicular lines. However, looking at the options, the first part: for statement 2, the reason is "Definition of perpendicular lines (so \(\angle W\) is a right angle, \(m\angle W = 90^\circ\))", but in the dropdown, the options include "Definition of perpendicular lines" (not listed exactly, but the key is that when two lines are perpendicular, the angle between them is \(90^\circ\)). Wait, maybe the user is asking for the reason for statement 2. Let's re - examine:

Given \(\overline{UW} \perp \overline{VW}\), by the definition of perpendicular lines, the angle between them (\(\angle W\)) is a right angle, and a right angle has a measure of \(90^\circ\). So the reason for \(m\angle W = 90^\circ\) is "Definition of perpendicular lines (or definition of a right angle, since perpendicular lines form a right angle)". But among the provided options in the dropdown (even though some are cut off), the correct reason for statement 2 is that perpendicular lines form a right angle, so \(m\angle W = 90^\circ\) (i.e., the definition of perpendicular lines implies \(m\angle W = 90^\circ\)).

Step2: Analyze the Proof Flow

- Statement 1: Given \(\overline{UW} \perp \overline{VW}\).
- Statement 2: \(m\angle W = 90^\circ\) (reason: definition of perpendicular lines, as perpendicular lines intersect at \(90^\circ\)).
- Statement 3: \(m\angle U + m\angle V + m\angle W=180^\circ\) (reason: Triangle Angle - Sum Theorem, which is the Additive Property of Angle Measure for a triangle's interior angles).
- Statement 4: Substitute \(m\angle W = 90^\circ\) into the equation from statement 3, so \(m\angle U + m\angle V+90^\circ = 180^\circ\) (reason: Substitution Property, using the value from statement 2 in statement 3).
- Statement 5: Subtract \(90^\circ\) from both sides of the equation in statement 4, so \(m\angle U + m\angle V=90^\circ\) (reason: Subtraction Property of Equality).

But focusing on statement 2, the reason for \(m\angle W = 90^\circ\) when \(\overline{UW} \perp \overline{VW}\) is the definition of perpendicular lines (two lines are perpendicular if they intersect at a right angle, and a right angle has measure \(90^\circ\)). Among the given options in the dropdown (even if some are partially visible), the correct reason for statement 2 is related to the definition of perpendicular lines leading to a right angle.

Answer:

For statement 2 (\(m\angle W = 90^\circ\)) the reason is "Definition of perpendicular lines (since \(\overline{UW} \perp \overline{VW}\), they form a right angle, so \(m\angle W = 90^\circ\))" (or from the given options, the appropriate one related to perpendicular lines forming a right angle). If we consider the options, the correct reason for \(m\angle W = 90^\circ\) given \(\overline{UW} \perp \overline{VW}\) is that perpendicular lines intersect at a right angle, so \(m\angle W = 90^\circ\) (by definition of perpendicular lines).