Question

critical thinking tell whether the statement is always, sometimes, or never true.

always sometimes never
vertical angles are supplements of each other.

explain your reasoning.

vertical angles are always congruent, so they will always be supplements of each other.
vertical angles are supplements of each other only when they are both 45°.
vertical angles are supplements of each other only when their rays form perpendicular lines.
vertical angles are congruent, so they are not supplements of each other.

Explanation:

Step1: Recall Vertical Angles Property

Vertical angles are always congruent (equal in measure). Let each vertical angle have measure \( x \). For them to be supplementary, \( x + x = 180^\circ \), so \( 2x = 180^\circ \) or \( x = 90^\circ \). This happens when the lines forming the vertical angles are perpendicular (since perpendicular lines form right angles, and vertical angles in that case are \( 90^\circ \), so \( 90^\circ+90^\circ = 180^\circ \), making them supplementary). For non - perpendicular lines, vertical angles are equal but not supplementary (e.g., if vertical angles are \( 30^\circ \), \( 30^\circ+30^\circ=60^\circ
eq180^\circ \)). So vertical angles are supplementary only sometimes (when the lines are perpendicular).

Step2: Analyze the Reasoning Options

• Option 1: Says vertical angles (always congruent) will always be supplementary. But as we saw, only when \( x = 90^\circ \) they are supplementary, so this is wrong.
• Option 2: Just restates the wrong idea, no valid reasoning.
• Option 3: Says they are supplementary only when both are \( 45^\circ \), but \( 45^\circ+45^\circ = 90^\circ

eq180^\circ \), so this is wrong.

• Option 4: Says vertical angles are supplementary only when their rays form perpendicular lines. When lines are perpendicular, vertical angles are \( 90^\circ \), so \( 90^\circ+90^\circ = 180^\circ \) (supplementary). When lines are not perpendicular, vertical angles are equal but not supplementary. So this reasoning is correct.

Answer:

For the first part (whether the statement "Vertical angles are supplements of each other" is always, sometimes, or never true): Sometimes.

For the reasoning: The correct reasoning option is "Vertical angles are supplements of each other only when their rays form perpendicular lines."