Question

in the figure below, ( mangle2 = 102^{circ} ). find ( mangle1 ), ( mangle3 ), and ( mangle4 ).

Explanation:

Step1: Find \( m\angle1 \) (supplementary angles)

\(\angle1\) and \(\angle2\) are supplementary, so \( m\angle1 + m\angle2 = 180^\circ \). Given \( m\angle2 = 102^\circ \), then \( m\angle1 = 180^\circ - 102^\circ = 78^\circ \).

Step2: Find \( m\angle3 \) (vertical angles)

\(\angle3\) and \(\angle1\) are vertical angles? Wait, no. \(\angle2\) and \(\angle4\) are vertical, \(\angle1\) and \(\angle3\)? Wait, no. Wait, \(\angle1\) and \(\angle3\) are supplementary? No, wait, \(\angle2\) and \(\angle1\) are supplementary (linear pair), \(\angle2\) and \(\angle3\) are supplementary? Wait, no, the figure is two intersecting lines, so \(\angle1\) and \(\angle2\) form a linear pair (sum to 180), \(\angle2\) and \(\angle3\) form a linear pair? Wait, no, let's correct. When two lines intersect, adjacent angles are supplementary (linear pair), and vertical angles are equal. So \(\angle1\) and \(\angle2\) are adjacent (linear pair), so \( m\angle1 + m\angle2 = 180^\circ \), so \( m\angle1 = 180 - 102 = 78^\circ \). Then \(\angle3\) is vertical to \(\angle1\)? No, wait, \(\angle2\) and \(\angle4\) are vertical angles (equal), \(\angle1\) and \(\angle3\) are vertical angles (equal). Wait, no, let's label: the two lines intersect, so angle 1 and angle 3 are vertical? Wait, no, angle 2 and angle 4 are vertical, angle 1 and angle 3 are vertical. Wait, no, in the figure, angle 2 is adjacent to angle 1 (linear pair), angle 2 is adjacent to angle 3 (linear pair), so angle 1 and angle 3 are equal (vertical angles)? Wait, no, if angle 2 and angle 1 are supplementary, angle 2 and angle 3 are supplementary, then angle 1 = angle 3 (since both are 180 - angle 2). Similarly, angle 2 and angle 4 are vertical (equal), so \( m\angle4 = m\angle2 = 102^\circ \), and \( m\angle3 = m\angle1 = 78^\circ \). Wait, I think I messed up earlier. Let's re-express:

• Linear pair: \( m\angle1 + m\angle2 = 180^\circ \) ⇒ \( m\angle1 = 180 - 102 = 78^\circ \)
• Vertical angles: \( m\angle3 = m\angle1 = 78^\circ \) (since they are vertical angles)
• Vertical angles: \( m\angle4 = m\angle2 = 102^\circ \) (since they are vertical angles)

Wait, that makes sense. So step 2: \( m\angle3 = m\angle1 = 78^\circ \) (vertical angles are equal). Step 3: \( m\angle4 = m\angle2 = 102^\circ \) (vertical angles are equal).

Answer:

\( m\angle1 = 78^\circ \)
\( m\angle3 = 78^\circ \)
\( m\angle4 = 102^\circ \)