Question - Solving for Angular Measures of Veins in a Leaf
Solution:
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.
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