Question

if \\(\overrightarrow{uv}\\) is an angle bisector of \\(\angle wvz\\) and \\(b = 30^\circ\\) (si \\(\overrightarrow{uv}\\) es una bisectriz de ángulo de \\(\angle wvz\\) y \\(b = 30^\circ\\)) (put your first name in the assignment box to get credit, ponga su nombre en el cuadro de la tarea para obtener crédito):
diagram with points w, u, z, y, x, v and angles a, b, c, d, e
what is the value of (¿cuál es el valor de):

• b?
• c?
• d?
• a?
• e?

Explanation:

For \( b \):

Step1: Given \( b = 30^\circ \)

\( b = 30^\circ \)

Step1: Angle bisector means \( b = c \)

Since \( \overrightarrow{UV} \) bisects \( \angle WVZ \), \( c = b \)

Step2: Substitute \( b = 30^\circ \)

\( c = 30^\circ \)

Step1: \( a \) is right angle (perpendicular lines)

Lines \( YZ \) and \( WX \) (assuming) are perpendicular? Wait, no, \( \angle a \) is between \( YV \) and \( WV \), but actually, looking at the diagram, \( \angle a \) is a right angle? Wait, no, \( \angle a + b + c = 180^\circ \)? Wait, no, \( YZ \) is a straight line, so \( \angle a + b + c = 180^\circ \)? Wait, no, \( YVZ \) is a straight line (180°), and \( UV \) is a bisector. Wait, actually, \( \angle a \) is 90°? Wait, no, let's re - examine. If \( YZ \) and \( WX \) are intersecting at \( V \), and \( UV \) is a bisector. Wait, maybe \( \angle a \) is 90°? No, wait, the sum of angles on a straight line is 180°. \( \angle a + b + c=180^\circ \)? But if \( b = c = 30^\circ \), then \( \angle a=180 - 30 - 30 = 120^\circ \)? Wait, no, maybe the lines are perpendicular. Wait, the diagram shows \( YVZ \) as a straight line (180°), and \( WXV \) as another straight line (180°). So \( \angle a \) and \( \angle d \) are vertical angles, \( \angle e \) and \( \angle c + b \) are vertical angles? Wait, no, let's start over.

Since \( \overrightarrow{UV} \) bisects \( \angle WVZ \), so \( \angle WVU=\angle UVZ = b = c = 30^\circ \). Now, \( YVZ \) is a straight line, so \( \angle YVW+\angle WVU+\angle UVZ = 180^\circ \). Let \( \angle YVW=a \), then \( a + b + c=180^\circ \). Substituting \( b = c = 30^\circ \), we get \( a+30 + 30=180 \), so \( a = 180 - 60=120^\circ \)? Wait, no, maybe the lines \( YZ \) and \( WX \) are perpendicular. Wait, the diagram has \( U \) at the top, \( Y \) left, \( Z \) right, \( W \) top - left, \( X \) bottom - right. So \( YVZ \) is horizontal, \( WVX \) is a line from top - left to bottom - right? No, \( W \) is top - left, \( X \) is bottom - right, so \( WVX \) is a straight line? Then \( \angle a \) and \( \angle d \) are vertical angles, \( \angle e \) and \( \angle c + b \) are vertical angles. Wait, if \( YVZ \) is horizontal (180°) and \( WVX \) is a line intersecting at \( V \), then \( \angle a+\angle b+\angle c = 180^\circ \) (since they are on a straight line \( YZ \)). We know \( b = c = 30^\circ \), so \( a=180-(30 + 30)=120^\circ \)? But maybe I made a mistake. Wait, the problem says "put your first name...", but for the angle \( a \):

Step1: Sum of angles on straight line

Angles on a straight line (\( YZ \)) sum to \( 180^\circ \), so \( a + b + c=180^\circ \)

Step2: Substitute \( b = 30^\circ \), \( c = 30^\circ \)

\( a+30 + 30=180 \)

Step3: Solve for \( a \)

\( a=180 - 60=120^\circ \)

Answer:

\( 30^\circ \)

For \( c \):