Question

1. ( l parallel m )

( mangle1 = ) ______
( mangle2 = ) ______
( mangle3 = ) ______
( mangle4 = ) ______
( mangle5 = ) ______
11.
( mangle1 = ) ____ ( mangle5 = ) ____
( mangle2 = ) ____ ( mangle6 = ) ____
( mangle3 = ) ____ ( mangle7 = ) ____
( mangle4 = ) ____ ( mangle8 = ) ____

Explanation:

Step1: Solve Problem 9 (Lines \( l \parallel m \))

\( m\angle1 \):

Since \( l \parallel m \), \( \angle1 \) and the \( 73^\circ \) angle are alternate interior angles. So \( m\angle1 = 73^\circ \).

\( m\angle2 \):

In the triangle, the sum of angles is \( 180^\circ \). We know two angles: \( 49^\circ \) and \( 73^\circ \). So \( m\angle2 = 180^\circ - 49^\circ - 73^\circ = 58^\circ \).

\( m\angle3 \):

\( \angle3 \) and the \( 73^\circ \) angle are supplementary (linear pair). So \( m\angle3 = 180^\circ - 73^\circ = 107^\circ \).

\( m\angle4 \):

\( \angle4 \) and the \( 49^\circ \) angle are alternate interior angles (since \( l \parallel m \)), so \( m\angle4 = 49^\circ \).

\( m\angle5 \):

\( \angle5 \) and \( \angle4 \) are supplementary (linear pair), so \( m\angle5 = 180^\circ - 49^\circ = 131^\circ \).

Step2: Solve Problem 11 (Triangle and Vertical Angles)

\( m\angle1 \):

In the triangle with \( 47^\circ \) and \( 42^\circ \), \( m\angle1 = 180^\circ - 47^\circ - 42^\circ = 91^\circ \). Wait, no—wait, vertical angles and triangle sum. Wait, \( \angle2 \) and the \( 71^\circ \) angle are vertical? No, \( \angle2 \) and \( 71^\circ \): Wait, \( \angle2 \) and the angle with \( 71^\circ \) are vertical? Wait, maybe better:

\( m\angle2 \):

\( \angle2 \) and \( 71^\circ \) are vertical angles? No, wait, \( \angle2 \) and the angle adjacent to \( 71^\circ \)? Wait, no, vertical angles are equal. Wait, \( \angle2 \) and the angle with \( 71^\circ \): Wait, maybe \( \angle2 = 71^\circ \) (vertical angles)? Wait, no, let's re-examine.

Wait, maybe I made a mistake. Let's start over for Problem 11:

\( m\angle1 \):

In the triangle with \( 47^\circ \) and \( 42^\circ \), \( m\angle1 = 180 - 47 - 42 = 91^\circ \)? No, wait, the triangle has angles \( 47^\circ \), \( 42^\circ \), and \( \angle1 \). So \( 47 + 42 + \angle1 = 180 \), so \( \angle1 = 91^\circ \).

\( m\angle2 \):

\( \angle2 \) and the \( 71^\circ \) angle: Wait, \( \angle2 \) and the angle with \( 71^\circ \) are vertical? No, \( \angle2 \) and \( 71^\circ \) are vertical? Wait, no, \( \angle2 \) and the angle opposite (let's say \( \angle5 \) adjacent? Wait, maybe \( \angle2 = 180 - 71 = 109^\circ \)? No, vertical angles: if \( \angle2 \) and \( 71^\circ \) are vertical, no. Wait, maybe \( \angle2 \) is vertical to an angle, but let's check the triangle.

Wait, maybe I messed up. Let's do Problem 9 first correctly:

Problem 9:
- \( l \parallel m \), so alternate interior angles: \( \angle1 = 73^\circ \) (alternate to \( 73^\circ \)).
- Triangle angles: \( 49^\circ + 73^\circ + \angle2 = 180^\circ \) ⇒ \( \angle2 = 180 - 49 - 73 = 58^\circ \).
- \( \angle3 \) is supplementary to \( 73^\circ \): \( 180 - 73 = 107^\circ \).
- \( \angle4 \) is alternate to \( 49^\circ \): \( 49^\circ \).
- \( \angle5 \) is supplementary to \( \angle4 \): \( 180 - 49 = 131^\circ \).

Problem 11:
- \( \angle1 \): In the triangle with \( 47^\circ \) and \( 42^\circ \), \( 47 + 42 + \angle1 = 180 \) ⇒ \( \angle1 = 91^\circ \).
- \( \angle2 \): Vertical to the angle with \( 71^\circ \)? Wait, no, \( \angle2 \) and \( 71^\circ \) are vertical? Wait, \( \angle2 \) and the angle opposite (let's say \( \angle5 \) adjacent) – no, \( \angle2 \) is vertical to an angle, but maybe \( \angle2 = 180 - 71 = 109^\circ \)? No, vertical angles are equal. Wait, maybe \( \angle2 = 71^\circ \) (vertical to the \( 71^\circ \) angle). Wait, the diagram shows \( 71^\circ \) and \( \angle2 \) as vertical? So \( m\angle2 = 71^\circ \).
- \( \angle3 \): Vertical to \( \angle5 \), but \( \angle3 = 180 - \angle2 = 180 - 71 = 109^\circ \)? No, \( \angle3 \) and \( \angle2 \) are supplementary? Wait, linear pair. If \( \angle2 = 71^\circ \), then \( \angle3 = 180 - 71 = 109^\circ \).
- \( \angle4 \): In the triangle with \( 36^\circ \) and \( 47^\circ \)? Wait, \( 36 + 47 + \angle4 = 180 \) ⇒ \( \angle4 = 180 - 36 - 47 = 97^\circ \)? No, maybe not. Wait, the diagram has a \( 36^\circ \) angle, \( 47^\circ \), and \( \angle4 \). So \( 36 + 47 + \angle4 = 180 \) ⇒ \( \angle4 = 97^\circ \).
- \( \angle5 \): Vertical to \( \angle2 \), so \( m\angle5 = 71^\circ \).
- \( \angle6 \): Alternate? Wait, \( \angle6 \) and \( 42^\circ \)? No, \( \angle6 = 42^\circ \) (alternate interior? Or vertical? Wait, \( \angle6 = 42^\circ \).
- \( \angle7 \): In the triangle with \( \angle6 \) and \( \angle1 \)? Wait, \( \angle7 = 180 - 91 - 42 = 47^\circ \)? No, maybe \( \angle7 = 47^\circ \) (alternate or vertical).
- \( \angle8 \): Supplementary to \( 36^\circ \) and \( \angle7 \)? Wait, \( 36 + 47 + \angle8 = 180 \) ⇒ \( \angle8 = 97^\circ \) (same as \( \angle4 \)).

But maybe I made errors. Let's focus on Problem 9 first:

Answer (Problem 9):

\( m\angle1 = 73^\circ \)
\( m\angle2 = 58^\circ \)
\( m\angle3 = 107^\circ \)
\( m\angle4 = 49^\circ \)
\( m\angle5 = 131^\circ \)

(Problem 11 needs more precise analysis, but let's confirm Problem 9 first.)

Answer:

(Problem 9):
\( m\angle1 = 73^\circ \)
\( m\angle2 = 58^\circ \)
\( m\angle3 = 107^\circ \)
\( m\angle4 = 49^\circ \)
\( m\angle5 = 131^\circ \)

(Problem 11 needs more precise analysis, but let's confirm Problem 9 first.)