Part A
To find the value of \( x \) when angle \( DEF \) and angle \( GEH \) are complementary, we first recall that complementary angles add up to \( 90^\circ \). Let's express the angles in terms of \( x \):
- Let \( \angle DEF = 2x + 10 \)
- Let \( \angle GEH = 3x - 20 \)
Since these angles are complementary, we can set up the following equation:
\[ \angle DEF + \angle GEH = 90^\circ \]
Substituting the expressions for the angles:
\[ (2x + 10) + (3x - 20) = 90 \]
Now combine like terms:
\[ 2x + 3x + 10 - 20 = 90 \] \[ 5x - 10 = 90 \]
Next, add \( 10 \) to both sides:
\[ 5x = 100 \]
Now, divide by \( 5 \):
\[ x = 20 \]
Part B
Now that we have the value of \( x \), we can find the measures of angles \( DEF \) and \( GEH \).
-
Find \( \angle DEF \): \[ \angle DEF = 2x + 10 = 2(20) + 10 = 40 + 10 = 50^\circ \]
-
Find \( \angle GEH \): \[ \angle GEH = 3x - 20 = 3(20) - 20 = 60 - 20 = 40^\circ \]
Thus, the measures of the angles are:
- \( \angle DEF = 50^\circ \)
- \( \angle GEH = 40^\circ \)
Part C
To determine whether angles \( DEF \) and \( GEH \) could also be vertical angles, we first need to recall the definition of vertical angles. Vertical angles are formed when two lines intersect, resulting in two pairs of opposite angles that are congruent.
Since angles \( DEF \) and \( GEH \) are complementary (their measures add up to \( 90^\circ \)), they cannot be vertical angles because vertical angles are always equal (congruent). Therefore, if \( DEF \) is \( 50^\circ \) and \( GEH \) is \( 40^\circ \), they cannot have the same measure necessary for them to be vertical angles.
Conclusion: No, angles \( DEF \) and \( GEH \) cannot be vertical angles because they are not equal; they measure \( 50^\circ \) and \( 40^\circ \) respectively.