To find the relationship between the angles given in the problem, let's first outline the statements:
- \( m\angle 1 = x + 72^\circ \)
- \( m\angle 2 = 2x \)
Part (a)
To find an equation for \( x \), we can assume that angles \( \angle 1 \) and \( \angle 2 \) are supplementary, which means their measures add up to \( 180^\circ \). Thus, we can write the equation:
\[ m\angle 1 + m\angle 2 = 180^\circ \]
Using the expressions for \( m\angle 1 \) and \( m\angle 2 \):
\[ (x + 72^\circ) + (2x) = 180^\circ \]
Now, combine like terms:
\[ 3x + 72^\circ = 180^\circ \]
Therefore, the equation to find \( x \) is:
\[ 3x + 72^\circ = 180^\circ \]
Part (b)
Now, we can solve for \( x \):
-
Subtract \( 72^\circ \) from both sides:
\[ 3x = 180^\circ - 72^\circ \]
\[ 3x = 108^\circ \]
-
Divide by 3:
\[ x = \frac{108^\circ}{3} \]
\[ x = 36^\circ \]
Now, we can find the measures of each angle.
For \( m\angle 1 \):
\[ m\angle 1 = x + 72^\circ = 36^\circ + 72^\circ = 108^\circ \]
For \( m\angle 2 \):
\[ m\angle 2 = 2x = 2(36^\circ) = 72^\circ \]
Final Answers
- Equation: \( 3x + 72^\circ = 180^\circ \)
- \( m\angle 1 = 108^\circ \)
- \( m\angle 2 = 72^\circ \)