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Answer:
To solve for \( m \) (angle measures), let's interpret the given information and solve step by step:
Given:
- \( m \angle DLS = 72^\circ \)
- \( m \angle DUP = 72^\circ \)
- \( m \angle DUP = 2x + 15 \)
- \( m \angle SUP = 7x - 6 \)
First, equate \( m \angle DUP \) to \( 72^\circ \):
\[ 2x + 15 = 72 \]
Subtract 15 from both sides:
\[ 2x = 72 - 15 \]
\[ 2x = 57 \]
Divide both sides by 2 to solve for \( x \):
\[ x = \frac{57}{2} \]
\[ x = 28.5 \]
Now that we have \( x \), substitute it back into \( m \angle SUP \) to find \( m \angle SUP \):
\[ m \angle SUP = 7x - 6 \]
\[ m \angle SUP = 7(28.5) - 6 \]
\[ m \angle SUP = 199.5 - 6 \]
\[ m \angle SUP = 193.5 \]
So, \( m \angle SUP = 193.5^\circ \).
Next, substitute \( x \) into \( m \angle DUP \) to find \( m \angle DUP \):
\[ m \angle DUP = 2x + 15 \]
\[ m \angle DUP = 2(28.5) + 15 \]
\[ m \angle DUP = 57 + 15 \]
\[ m \angle DUP = 72 \]
So, \( m \angle DUP = 72^\circ \).
Therefore, the angle measures are:
- \( m \angle DLS = 72^\circ \)
- \( m \angle DUP = 72^\circ \)
- \( m \angle SUP = 193.5^\circ \)