Question

in the figure to the right, s||t. use the figure to answer the following exercise. ∠c is congruent to which two of the following angles? select all that apply. □ ∠a □ ∠b □ ∠e □ ∠f

Explanation:

Step1: Recall Parallel Line Angle Rules

When two parallel lines (\(s \parallel t\)) are cut by a transversal (line \(u\)), we use rules like alternate interior angles, vertical angles, and corresponding angles. Vertical angles are congruent, alternate interior angles are congruent, and corresponding angles are congruent.

Step2: Analyze \(\angle c\) and \(\angle a\)

\(\angle a\) and \(\angle c\): Since \(s \parallel t\) and \(u\) is a transversal, \(\angle a\) and \(\angle c\) are alternate interior angles. Alternate interior angles are congruent, so \(\angle c \cong \angle a\).

Step3: Analyze \(\angle c\) and \(\angle f\)

\(\angle f\) and \(\angle a\) are vertical angles (so \(\angle f \cong \angle a\)). Since \(\angle c \cong \angle a\), by transitivity, \(\angle c \cong \angle f\). Also, \(\angle f\) and \(\angle c\) can be seen as corresponding or via vertical/alternate angles. Wait, also \(\angle b\) and \(\angle d\) etc., but let's check each option:

• \(\angle a\): Alternate interior, congruent.
• \(\angle f\): \(\angle f\) and \(\angle a\) are vertical (congruent), so \(\angle f \cong \angle c\) (since \(\angle c \cong \angle a\)).
• \(\angle b\): \(\angle b\) and \(\angle d\) are alternate, but \(\angle b\) and \(\angle c\): \(\angle b\) and \(\angle a\) are supplementary? Wait no, \(\angle a\) and \(\angle b\) are adjacent supplementary, but \(\angle c\) and \(\angle a\) are congruent, so \(\angle c\) and \(\angle b\) are supplementary? Wait maybe I made a mistake. Wait, let's re-examine:

Wait, line \(s\) and \(t\) are parallel, transversal \(u\). \(\angle a\) and \(\angle c\): alternate interior (between \(s\) and \(t\), inside, on opposite sides of transversal) – yes, so congruent. \(\angle f\) and \(\angle a\) are vertical angles (formed by intersection of \(s\) and \(u\)), so \(\angle f = \angle a\), so \(\angle f = \angle c\) (since \(\angle a = \angle c\)). \(\angle e\) is supplementary to \(\angle f\), so \(\angle e\) is supplementary to \(\angle c\), not congruent. \(\angle b\): \(\angle b\) and \(\angle a\) are supplementary (linear pair), so \(\angle b\) is supplementary to \(\angle c\) (since \(\angle a = \angle c\)), so not congruent. Wait, but maybe I messed up. Wait, \(\angle b\) and \(\angle d\) are alternate interior? No, \(\angle b\) and \(\angle d\): \(\angle b\) is on line \(s\), \(\angle d\) on line \(t\), transversal \(u\) – they are corresponding angles? Wait, no, \(\angle b\) and \(\angle d\) are same - side? Wait, maybe the correct angles are \(\angle a\) and \(\angle f\). Wait, let's check again:

Vertical angles: \(\angle f\) and \(\angle e\) are supplementary, \(\angle f\) and \(\angle a\) are vertical (so \(\angle f = \angle a\)). Alternate interior: \(\angle a\) and \(\angle c\) (since \(s \parallel t\), transversal \(u\)) – so \(\angle a = \angle c\). Therefore, \(\angle c = \angle a\) (alternate interior) and \(\angle c = \angle f\) (since \(\angle f = \angle a\) (vertical) and \(\angle a = \angle c\) (alternate interior)). So the congruent angles to \(\angle c\) are \(\angle a\) and \(\angle f\).

Answer:

\(\boldsymbol{\angle a}\), \(\boldsymbol{\angle f}\) (so the checkboxes for \(\angle a\) and \(\angle f\) should be selected)