Question

21 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 the measuremen diagram with lines mn, pq, and transversal st, angles labeled 1 - 8, angle 6 is 128° m∠1 = 52 degrees m∠2 = 128 degrees m∠3 = 52 degrees

Explanation:

Step1: Analyze ∠1 and ∠6

Since \( \overleftrightarrow{MN} \parallel \overleftrightarrow{PQ} \), ∠1 and ∠6 are same - side interior angles? No, wait, ∠1 and ∠5? Wait, no, ∠6 and ∠2: Wait, ∠6 is 128°, and ∠1 and ∠6: Wait, actually, ∠1 and ∠5: Wait, no, let's look at the linear pair. ∠6 and ∠5 are supplementary? Wait, ∠6 is 128°, so ∠5 = 180 - 128 = 52°. Then, since \( \overleftrightarrow{MN} \parallel \overleftrightarrow{PQ} \), ∠1 and ∠5 are corresponding angles, so \( m\angle1 = 52^\circ \).

Step2: Analyze ∠2 and ∠6

∠2 and ∠6 are corresponding angles (since \( \overleftrightarrow{MN} \parallel \overleftrightarrow{PQ} \)), so \( m\angle2=m\angle6 = 128^\circ \).

Step3: Analyze ∠3 and ∠1

∠3 and ∠1 are vertical angles? No, ∠3 and ∠1: Wait, ∠3 and ∠2 are supplementary, so \( m\angle3=180 - m\angle2=180 - 128 = 52^\circ \). Or, ∠3 and ∠1: ∠3 and ∠1 are corresponding to ∠7 and ∠5? Wait, simpler: ∠3 and ∠1, since ∠1 is 52°, and ∠3 is vertical to ∠1? No, ∠3 and ∠1: Wait, ∠3 and ∠2 are supplementary, so \( m\angle3 = 180 - 128=52^\circ \).

Answer:

\( m\angle1 = 52 \) degrees, \( m\angle2 = 128 \) degrees, \( m\angle3 = 52 \) degrees.