Question

gardening a gardener uses a grow light to grow vegetables indoors. if $m\angle1 = (8x)^\circ$ and $m\angle2 = (7x - 4)^\circ$, what is $m\angle1$?

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Explanation:

Step1: Identify angle relationship

From the diagram, we can see that \( \angle 1 \), \( \angle 2 \), and the \( 116^\circ \) angle form a triangle? Wait, no, actually, looking at the diagram, the two angles \( \angle 1 \) and the \( 116^\circ \) angle are supplementary? Wait, no, let's think again. Wait, the diagram shows that \( \angle 1 \) and the \( 116^\circ \) angle are adjacent? Wait, no, maybe the sum of \( \angle 1 \), \( \angle 2 \), and the \( 116^\circ \) angle? Wait, no, actually, in the diagram, the two angles \( \angle 1 \) and the \( 116^\circ \) angle are equal? Wait, no, the problem says \( m\angle 1=(8x)^\circ \) and \( m\angle 2=(7x - 4)^\circ \). Wait, maybe the triangle has angles \( \angle 1 \), \( \angle 2 \), and the angle adjacent to \( 116^\circ \). Wait, the \( 116^\circ \) angle and its adjacent angle are supplementary, so the adjacent angle is \( 180 - 116=64^\circ \). Then, in the triangle, the sum of angles is \( 180^\circ \), so \( \angle 1+\angle 2 + 64^\circ=180^\circ \). So \( 8x+(7x - 4)+64 = 180 \).

Step2: Solve for x

Combine like terms: \( 8x+7x-4 + 64=180 \)
\( 15x+60 = 180 \)
Subtract 60 from both sides: \( 15x=180 - 60=120 \)
Divide both sides by 15: \( x=\frac{120}{15}=8 \)

Step3: Find \( m\angle 1 \)

Since \( m\angle 1=(8x)^\circ \), substitute \( x = 8 \): \( 8\times8 = 64^\circ \)

Wait, but let's check again. Wait, if \( x = 8 \), then \( m\angle 2=(7\times8 - 4)=56 - 4 = 52^\circ \). Then \( 64+52+64 = 180 \)? Wait, no, \( 64+52+64 = 180 \)? \( 64+52=116 \), \( 116+64 = 180 \). Yes, that works. So \( m\angle 1 = 64^\circ \)? Wait, but wait, the \( 116^\circ \) angle's adjacent angle is \( 64^\circ \), so the triangle has angles \( 64^\circ \) (adjacent to \( 116^\circ \)), \( \angle 1 = 64^\circ \), and \( \angle 2 = 52^\circ \). Wait, but that would mean \( \angle 1 \) is equal to the adjacent angle of \( 116^\circ \). Maybe the diagram shows that \( \angle 1 \) and the angle adjacent to \( 116^\circ \) are equal, so \( \angle 1 = 64^\circ \), and then we can solve for \( x \) and \( \angle 2 \).

Wait, let's re - do the steps:

1. The angle adjacent to \( 116^\circ \) is \( 180 - 116=64^\circ \) (linear pair).
2. In the triangle, the sum of interior angles is \( 180^\circ \), so \( \angle 1+\angle 2+64^\circ = 180^\circ \).
3. Substitute \( \angle 1 = 8x \) and \( \angle 2=7x - 4 \):

\( 8x+(7x - 4)+64 = 180 \)
\( 15x+60 = 180 \)
\( 15x=120 \)
\( x = 8 \)

1. Then \( \angle 1=8x=8\times8 = 64^\circ \)

Answer:

\( 64^\circ \)