Question

in the diagram, \overleftrightarrow{mn} is parallel to \overleftrightarrow{pq}.
find the following angle measures.
hint: draw this transversal on your paper, label the angles and th
diagram with lines mn, pq, and transversal st, angles labeled 1 - 8, angle 6 is 128°

Explanation:

Step1: Identify Angle Relationships

Since \( \overleftrightarrow{MN} \parallel \overleftrightarrow{PQ} \) and \( \overleftrightarrow{ST} \) is a transversal, we use properties of parallel lines (corresponding angles, vertical angles, linear pairs).

Step2: Find \( m\angle 1 \)

\( \angle 6 = 128^\circ \), \( \angle 1 \) and \( \angle 6 \) are same - side interior angles? Wait, no. Wait, \( \angle 2 \) and \( \angle 6 \): Wait, \( \angle 2 \) and \( \angle 6 \) are corresponding angles? Wait, no, let's look at linear pairs. \( \angle 6 \) and \( \angle 7 \) are linear pair? Wait, \( \angle 6 = 128^\circ \), so \( \angle 5 \) and \( \angle 6 \) are linear pair? Wait, \( \angle 5 + \angle 6=180^\circ \), so \( \angle 5 = 180 - 128=52^\circ \). Then \( \angle 1 \) and \( \angle 5 \) are corresponding angles (since \( MN\parallel PQ \)), so \( \angle 1=\angle 5 = 52^\circ \)? Wait, no, maybe \( \angle 1 \) and \( \angle 6 \): Wait, \( \angle 1 \) and \( \angle 2 \) are vertical angles? Wait, no, let's re - examine.

Wait, \( \angle 2 \) and \( \angle 6 \): Since \( MN\parallel PQ \), \( \angle 2 \) and \( \angle 6 \) are corresponding angles? Wait, \( \angle 6 = 128^\circ \), so \( \angle 2 = 128^\circ \). Then \( \angle 1 \) and \( \angle 2 \) are linear pair, so \( \angle 1=180 - 128 = 52^\circ \).

Step3: Find \( m\angle 2 \)

\( \angle 2 \) and \( \angle 6 \) are corresponding angles (because \( MN\parallel PQ \) and \( ST \) is transversal), so \( m\angle 2=m\angle 6 = 128^\circ \).

Step4: Find \( m\angle 3 \)

\( \angle 3 \) and \( \angle 1 \) are vertical angles? Wait, \( \angle 3 \) and \( \angle 1 \): No, \( \angle 3 \) and \( \angle 2 \) are vertical angles? Wait, \( \angle 3 \) and \( \angle 1 \): Wait, \( \angle 3 \) and \( \angle 2 \) are linear pair? No, \( \angle 3 \) and \( \angle 4 \) are vertical angles? Wait, \( \angle 3 \) and \( \angle 1 \): Wait, \( \angle 3 \) and \( \angle 2 \) are adjacent, \( \angle 3+\angle 2 = 180^\circ \)? No, \( \angle 3 \) and \( \angle 4 \) are vertical angles, \( \angle 2 \) and \( \angle 4 \) are vertical angles? Wait, no, let's use vertical angles. \( \angle 3 \) and \( \angle 1 \) are vertical angles? Wait, \( \angle 3 \) and \( \angle 1 \): If \( \angle 1 = 52^\circ \), then \( \angle 3=\angle 1 = 52^\circ \)? No, wait \( \angle 3 \) and \( \angle 2 \) are vertical angles? Wait, \( \angle 2 = 128^\circ \), so \( \angle 3=\angle 2 = 128^\circ \) (vertical angles).

Step5: Find \( m\angle 4 \)

\( \angle 4 \) and \( \angle 1 \) are vertical angles? Wait, \( \angle 4 \) and \( \angle 1 \): No, \( \angle 4 \) and \( \angle 2 \) are adjacent? Wait, \( \angle 4 \) and \( \angle 1 \) are vertical angles? Wait, \( \angle 4 \) and \( \angle 1 \): If \( \angle 1 = 52^\circ \), then \( \angle 4=\angle 1 = 52^\circ \) (vertical angles) or \( \angle 4 \) and \( \angle 3 \) are linear pair? Wait, \( \angle 3 = 128^\circ \), so \( \angle 4=180 - 128 = 52^\circ \).

Let's summarize:

• \( m\angle 1 = 52^\circ \) (linear pair with \( \angle 2 \), and \( \angle 2=\angle 6 = 128^\circ \))
• \( m\angle 2 = 128^\circ \) (corresponding angles with \( \angle 6 \))
• \( m\angle 3 = 128^\circ \) (vertical angles with \( \angle 2 \))
• \( m\angle 4 = 52^\circ \) (vertical angles with \( \angle 1 \))

(Assuming we are to find these angles, if the question was to find specific angles, for example, if we assume we need to find \( m\angle 1 \), \( m\angle 2 \), \( m\angle 3 \), \( m\angle 4 \))

Answer:

If we take \( \angle 1 \): \( \boldsymbol{52^\circ} \), \( \angle 2 \): \( \boldsymbol{128^\circ} \), \( \angle 3 \): \( \boldsymbol{128^\circ} \), \( \angle 4 \): \( \boldsymbol{52^\circ} \) (depending on which angle is asked, but following the steps above)