Question

which set of statements about the angles is true?
○ $\angle 1 \cong \angle 6, \angle 2 \cong \angle 4, \angle 3 \cong \angle 5$
○ $\angle 1 \cong \angle 5, \angle 2 \cong \angle 4, \angle 3 \cong \angle 6$
○ $\angle 6 \cong \angle 2, \angle 5 \cong \angle 4, \angle 1 \cong \angle 3$
○ $\angle 6 \cong \angle 1, \angle 3 \cong \angle 2, \angle 4 \cong \angle 5$

Explanation:

Step1: Identify Parallel Lines and Transversals

The vertical lines (with orange arrows) are parallel (let's call them \( l_1 \) and \( l_2 \)). The lines \( s \) and \( t \) are transversals intersecting these parallel lines, and the two slanted lines also intersect, forming vertical angles.

Step2: Analyze Vertical Angles

Vertical angles are equal. \( \angle 3 \) and \( \angle 6 \) are vertical angles, so \( \angle 3 \cong \angle 6 \). \( \angle 1 \) and \( \angle 5 \): Since \( l_1 \parallel l_2 \) and \( t \) is a transversal, \( \angle 1 \) and \( \angle 5 \) are corresponding angles, so \( \angle 1 \cong \angle 5 \). \( \angle 2 \) and \( \angle 4 \): Since \( l_1 \parallel l_2 \) and \( s \) is a transversal, \( \angle 2 \) and \( \angle 4 \) are corresponding angles, so \( \angle 2 \cong \angle 4 \).

Step3: Check Each Option

• First option: \( \angle 1 \cong \angle 6 \) (false), \( \angle 2 \cong \angle 4 \) (true), \( \angle 3 \cong \angle 5 \) (false). Eliminate.
• Second option: \( \angle 1 \cong \angle 5 \) (true), \( \angle 2 \cong \angle 4 \) (true), \( \angle 3 \cong \angle 6 \) (true). This matches.
• Third option: \( \angle 6 \cong \angle 2 \) (false), \( \angle 5 \cong \angle 4 \) (false), \( \angle 1 \cong \angle 3 \) (false). Eliminate.
• Fourth option: \( \angle 6 \cong \angle 1 \) (false), \( \angle 3 \cong \angle 2 \) (false), \( \angle 4 \cong \angle 5 \) (false, they are supplementary). Eliminate.

Answer:

\( \angle 1 \cong \angle 5, \angle 2 \cong \angle 4, \angle 3 \cong \angle 6 \) (the second option)