Question

multiple choice (18 points) if $\angle 2 = 95^\circ$, what is the value of $\angle 7$? si $\angle 2 = 95^\circ$, ¿cuál es el valor de $\angle 7$? $\overleftrightarrow{cf} \parallel \overleftrightarrow{jk}$ diagram shows two parallel lines $\overleftrightarrow{cf}$ (with points c, f) and $\overleftrightarrow{jk}$ (with points j, k) cut by a transversal, forming angles 1,2,3,4 on $\overleftrightarrow{cf}$ and 5,6,7,8 on $\overleftrightarrow{jk}$ options: 105, 5, 85, 95

Explanation:

Step1: Identify Relationship (Parallel Lines)

Since \( CF \parallel JK \) and a transversal intersects them, \( \angle 2 \) and \( \angle 6 \) are corresponding angles? Wait, no—wait, \( \angle 2 \) and \( \angle 6 \)? Wait, actually, \( \angle 2 \) and \( \angle 6 \) are corresponding? Wait, no, maybe \( \angle 2 \) and \( \angle 6 \) are equal? Wait, no, wait the diagram: \( CF \) and \( JK \) are parallel, transversal cuts them. \( \angle 2 \) and \( \angle 6 \) – no, wait \( \angle 2 \) and \( \angle 6 \) are corresponding? Wait, no, maybe \( \angle 2 \) and \( \angle 6 \) are equal? Wait, no, wait the options: one of them is 85? Wait, no, wait \( \angle 2 \) and \( \angle 3 \) are supplementary? Wait, no, \( \angle 2 \) and \( \angle 6 \): wait, \( CF \parallel JK \), so \( \angle 2 \) and \( \angle 6 \) are equal (corresponding angles)? Wait, no, maybe \( \angle 2 \) and \( \angle 6 \) – wait, the angle \( \angle 7 \)? Wait, the question is about \( \angle 7 \)? Wait, the original question: "If \( \angle 2 = 95^\circ \), what is the value of \( \angle 7 \)?" (assuming typo, maybe \( \angle 7 \)). Since \( CF \parallel JK \), \( \angle 2 \) and \( \angle 6 \) are equal (corresponding), and \( \angle 6 \) and \( \angle 7 \) are supplementary (linear pair). Wait, no: \( \angle 2 \) and \( \angle 3 \) are supplementary (linear pair: \( \angle 2 + \angle 3 = 180^\circ \)), and \( \angle 3 \) and \( \angle 7 \) are corresponding angles (since \( CF \parallel JK \)), so \( \angle 3 = \angle 7 \). So first, find \( \angle 3 \): \( \angle 3 = 180^\circ - \angle 2 = 180 - 95 = 85^\circ \). Then \( \angle 7 = \angle 3 = 85^\circ \)? Wait, no, maybe \( \angle 2 \) and \( \angle 7 \) are alternate exterior angles? Wait, let's re-examine.

Wait, \( CF \parallel JK \), transversal. \( \angle 2 \) and \( \angle 6 \) are corresponding (so \( \angle 6 = 95^\circ \)). Then \( \angle 6 \) and \( \angle 7 \) are linear pair, so \( \angle 6 + \angle 7 = 180^\circ \), so \( \angle 7 = 180 - 95 = 85^\circ \). Alternatively, \( \angle 2 \) and \( \angle 3 \) are linear pair (\( \angle 2 + \angle 3 = 180 \)), so \( \angle 3 = 85 \), and \( \angle 3 \) and \( \angle 7 \) are corresponding (since \( CF \parallel JK \)), so \( \angle 7 = 85 \). So the answer is 85.

Step1: Find supplementary angle to \( \angle 2 \)

\( \angle 2 + \angle 3 = 180^\circ \) (linear pair), so \( \angle 3 = 180^\circ - 95^\circ = 85^\circ \).

Step2: Corresponding angles (parallel lines)

Since \( CF \parallel JK \), \( \angle 3 = \angle 7 \) (corresponding angles), so \( \angle 7 = 85^\circ \).

Answer:

85