Question

thought provoking the diagram shows an intersection of roads. identify all supplementary, complementary, and vertical angles.
supplementary angles
complementary angles
vertical angles
options: ∠1 and ∠4, ∠1 and ∠2, ∠1 and ∠5, ∠2 and ∠5, ∠2 and ∠3, ∠4 and ∠5, ∠3 and ∠4

Explanation:

To solve this, we analyze each pair using angle definitions:

Supplementary Angles (sum to \(180^\circ\)):

• \(\angle 1\) and \(\angle 5\): \(\angle 1\) is a right angle? Wait, no—\(\angle 1\) and \(\angle 5\) are adjacent and form a linear pair? Wait, looking at the diagram, \(\angle 1\) and \(\angle 5\) are on a straight line (Genview St. and Plum Rd. intersection), so they are supplementary. Also, \(\angle 4\) and \(\angle 5\): \(\angle 4\) is a right angle? Wait, \(\angle 4\) and \(\angle 5\) are adjacent on a straight line? Wait, maybe \(\angle 1\) and \(\angle 4\)? No, let's re-express. Wait, the options given:
• \(\angle 1\) and \(\angle 5\): If \(\angle 1\) and \(\angle 5\) are adjacent and form a linear pair (sum \(180^\circ\)), they are supplementary.
• \(\angle 4\) and \(\angle 5\): Similarly, if they form a linear pair, supplementary.

Complementary Angles (sum to \(90^\circ\)):

• \(\angle 2\) and \(\angle 3\): If \(\angle 2 + \angle 3 = 90^\circ\) (since there’s a right angle implied in the diagram, like Plum Rd. and the vertical line), they are complementary.
• \(\angle 1\) and \(\angle 2\): If \(\angle 1\) is \(90^\circ\) (right angle) and \(\angle 1 + \angle 2 = 90^\circ\)? No, wait—if \(\angle 1\) is a right angle, then \(\angle 1 = 90^\circ\), so \(\angle 1 + \angle 2 = 90^\circ\) would mean \(\angle 2 = 0^\circ\), which is wrong. Wait, maybe \(\angle 2\) and \(\angle 3\) are complementary (sum \(90^\circ\)).

Vertical Angles (opposite, equal angles):

• \(\angle 1\) and \(\angle 4\): If \(\angle 1\) and \(\angle 4\) are vertical angles (formed by intersecting lines), they are equal and vertical.

Let’s match the options:

Supplementary Angles:

• \(\angle 1\) and \(\angle 5\) (linear pair, sum \(180^\circ\))
• \(\angle 4\) and \(\angle 5\) (linear pair, sum \(180^\circ\))

Complementary Angles:

• \(\angle 2\) and \(\angle 3\) (sum \(90^\circ\), as they are part of a right angle)

Vertical Angles:

• \(\angle 1\) and \(\angle 4\) (opposite angles from intersecting lines, equal)

Final Categorization (example for each category):

• Supplementary Angles: \(\boldsymbol{\angle 1}\) and \(\boldsymbol{\angle 5}\) (or \(\angle 4\) and \(\angle 5\))
• Complementary Angles: \(\boldsymbol{\angle 2}\) and \(\boldsymbol{\angle 3}\)
• Vertical Angles: \(\boldsymbol{\angle 1}\) and \(\boldsymbol{\angle 4}\)

(Note: The exact pairs depend on the diagram’s right angles and intersections, but using the given options, we assign as above.)

Answer:

To solve this, we analyze each pair using angle definitions:

Supplementary Angles (sum to \(180^\circ\)):

• \(\angle 1\) and \(\angle 5\): \(\angle 1\) is a right angle? Wait, no—\(\angle 1\) and \(\angle 5\) are adjacent and form a linear pair? Wait, looking at the diagram, \(\angle 1\) and \(\angle 5\) are on a straight line (Genview St. and Plum Rd. intersection), so they are supplementary. Also, \(\angle 4\) and \(\angle 5\): \(\angle 4\) is a right angle? Wait, \(\angle 4\) and \(\angle 5\) are adjacent on a straight line? Wait, maybe \(\angle 1\) and \(\angle 4\)? No, let's re-express. Wait, the options given:
• \(\angle 1\) and \(\angle 5\): If \(\angle 1\) and \(\angle 5\) are adjacent and form a linear pair (sum \(180^\circ\)), they are supplementary.
• \(\angle 4\) and \(\angle 5\): Similarly, if they form a linear pair, supplementary.

Complementary Angles (sum to \(90^\circ\)):

• \(\angle 2\) and \(\angle 3\): If \(\angle 2 + \angle 3 = 90^\circ\) (since there’s a right angle implied in the diagram, like Plum Rd. and the vertical line), they are complementary.
• \(\angle 1\) and \(\angle 2\): If \(\angle 1\) is \(90^\circ\) (right angle) and \(\angle 1 + \angle 2 = 90^\circ\)? No, wait—if \(\angle 1\) is a right angle, then \(\angle 1 = 90^\circ\), so \(\angle 1 + \angle 2 = 90^\circ\) would mean \(\angle 2 = 0^\circ\), which is wrong. Wait, maybe \(\angle 2\) and \(\angle 3\) are complementary (sum \(90^\circ\)).

Vertical Angles (opposite, equal angles):

• \(\angle 1\) and \(\angle 4\): If \(\angle 1\) and \(\angle 4\) are vertical angles (formed by intersecting lines), they are equal and vertical.

Let’s match the options:

Supplementary Angles:

• \(\angle 1\) and \(\angle 5\) (linear pair, sum \(180^\circ\))
• \(\angle 4\) and \(\angle 5\) (linear pair, sum \(180^\circ\))

Complementary Angles:

• \(\angle 2\) and \(\angle 3\) (sum \(90^\circ\), as they are part of a right angle)

Vertical Angles:

• \(\angle 1\) and \(\angle 4\) (opposite angles from intersecting lines, equal)

Final Categorization (example for each category):

• Supplementary Angles: \(\boldsymbol{\angle 1}\) and \(\boldsymbol{\angle 5}\) (or \(\angle 4\) and \(\angle 5\))
• Complementary Angles: \(\boldsymbol{\angle 2}\) and \(\boldsymbol{\angle 3}\)
• Vertical Angles: \(\boldsymbol{\angle 1}\) and \(\boldsymbol{\angle 4}\)

(Note: The exact pairs depend on the diagram’s right angles and intersections, but using the given options, we assign as above.)