Question

1. flags the alabama state flag is white and has two diagonal red stripes. if the ( mangle1 = 112^circ ), what is ( mangle2 )?
2. construction alan has installed a new window above the entrance of an office building. if ( mangle2 = 44^circ ), what is ( mangle1 )?

Explanation:

Problem 4: FLAGS

Step1: Identify angle relationship

The two red stripes form vertical angles and adjacent supplementary angles? Wait, no, in the flag, the two red diagonals form a cross, so ∠1 and ∠2 are vertical angles? Wait, no, looking at the diagram, ∠1 and ∠2 are adjacent and form a linear pair? Wait, no, the sum of ∠1 and ∠2? Wait, no, actually, in the flag, the two red diagonals intersect, so ∠1 and ∠2 are vertical angles? Wait, no, the measure of ∠1 is 112°, and we need to find ∠2. Wait, maybe they are supplementary? Wait, no, let's think again. Wait, the flag is a rectangle, and the two red diagonals form a cross. So ∠1 and ∠2 are vertical angles? No, wait, if ∠1 is 112°, then ∠2 should be equal to ∠1? No, that can't be. Wait, maybe they are supplementary. Wait, 180 - 112 = 68? Wait, no, maybe I made a mistake. Wait, the problem says "the Alabama state flag is white and has two diagonal red stripes. If m∠1 = 112°, what is m∠2?" Let's assume that ∠1 and ∠2 are adjacent and form a linear pair? No, linear pair sums to 180. Wait, 180 - 112 = 68? Wait, maybe the diagram shows that ∠1 and ∠2 are vertical angles? No, vertical angles are equal. Wait, maybe the two red stripes are parallel? No, they are diagonals. Wait, maybe the angle between them is supplementary. Wait, let's check the diagram. The diagram shows a white rectangle with two red diagonals crossing, forming four angles. ∠1 and ∠2 are adjacent. So if ∠1 is 112°, then ∠2 is 180 - 112 = 68°? Wait, that makes sense because they form a linear pair (adjacent angles on a straight line, sum to 180°).

Step2: Calculate m∠2

Since ∠1 and ∠2 form a linear pair, their measures sum to 180°. So \( m\angle 2 = 180^\circ - m\angle 1 \).
Substitute \( m\angle 1 = 112^\circ \):
\( m\angle 2 = 180^\circ - 112^\circ = 68^\circ \)

Step1: Identify angle relationship

In the window diagram, there is a right angle? Wait, the window has a semicircular top? Wait, the diagram shows a door with a semicircular window above it, and there are lines from the center. Let's assume that the horizontal line is a straight line, so the angle on the straight line is 180°, and there is a right angle? Wait, no, the problem says "Alan has installed a new window above the entrance of an office building. If m∠2 = 44°, what is m∠1?" Let's assume that ∠1 and ∠2 are complementary? Wait, no, maybe the total angle is 90°? Wait, looking at the diagram, there is a right angle (90°) formed? Wait, maybe the sum of ∠1 and ∠2 is 90°? Wait, if ∠2 is 44°, then ∠1 = 90° - 44° = 46°? Wait, no, maybe the total angle is 180°? Wait, no, the diagram shows a semicircular window with lines from the center, forming angles. Let's assume that the horizontal line is a straight line, and there is a right angle. Wait, maybe the angle between the two lines is 90°, so ∠1 + ∠2 = 90°? So \( m\angle 1 = 90^\circ - m\angle 2 \).

Step2: Calculate m∠1

Substitute \( m\angle 2 = 44^\circ \):
\( m\angle 1 = 90^\circ - 44^\circ = 46^\circ \)

Answer:

\( 68^\circ \)

Problem 5: CONSTRUCTION