Solving for Angular Measures of Veins in a Leaf
The problem states that the veins in a leaf form a pair of supplementary angles. Supplementary angles are two angles whose measures add up to 180 degrees. It gives us the measures of the two angles in terms of x:
\( m\angle1 = 7x + 13^\circ \)
\( m\angle2 = 25x + 7^\circ \)
Because they are supplementary, we can set up the following equation to solve for x:
\( 7x + 13^\circ + 25x + 7^\circ = 180^\circ \)
Combine like terms:
\( 32x + 20^\circ = 180^\circ \)
Subtract 20 degrees from both sides to isolate the terms with x:
\( 32x = 160^\circ \)
Divide both sides by 32 to find x:
\( x = 160^\circ / 32 = 5^\circ \)
Now that we know the value of x, we can substitute it into the expressions for the angles to find their measures:
\( m\angle1 = 7(5^\circ) + 13^\circ = 35^\circ + 13^\circ = 48^\circ \)
\( m\angle2 = 25(5^\circ) + 7^\circ = 125^\circ + 7^\circ = 132^\circ \)
Therefore, the measures of the angles are 48 degrees and 132 degrees, respectively.
