Sure, let's solve this step-by-step.
Step 1: Define the unknown angle.
Let the unknown angle be [tex]\( x \)[/tex].
Step 2: Express the supplementary and complementary angles in terms of [tex]\( x \)[/tex].
- The supplementary angle of [tex]\( x \)[/tex] is [tex]\( 180^\circ - x \)[/tex].
- The complementary angle of [tex]\( x \)[/tex] is [tex]\( 90^\circ - x \)[/tex].
Step 3: Set up the equation based on the problem statement.
According to the problem, the supplementary angle is 35 degrees more than twice the complementary angle. This gives us the equation:
[tex]\[ 180^\circ - x = 35^\circ + 2 \cdot (90^\circ - x) \][/tex]
Step 4: Simplify the equation.
First, simplify the expression on the right side of the equation:
[tex]\[ 180^\circ - x = 35^\circ + 2 \cdot (90^\circ - x) \][/tex]
[tex]\[ 180^\circ - x = 35^\circ + 180^\circ - 2x \][/tex]
Step 5: Combine like terms.
[tex]\[ 180^\circ - x = 215^\circ - 2x \][/tex]
Step 6: Isolate the variable [tex]\( x \)[/tex].
Add [tex]\( 2x \)[/tex] to both sides:
[tex]\[ 180^\circ + x = 215^\circ \][/tex]
Subtract [tex]\( 180^\circ \)[/tex] from both sides:
[tex]\[ x = 35^\circ \][/tex]
Step 7: Check the results.
- The unknown angle is [tex]\( x = 35^\circ \)[/tex].
- The supplementary angle is [tex]\( 180^\circ - 35^\circ = 145^\circ \)[/tex].
- The complementary angle is [tex]\( 90^\circ - 35^\circ = 55^\circ \)[/tex].
Now, let's verify:
The supplementary angle (145°) should be 35° more than twice the complementary angle:
[tex]\[ 2 \cdot 55^\circ + 35^\circ = 110^\circ + 35^\circ = 145^\circ \][/tex]
Since the calculations are consistent with the problem statement:
- The unknown angle is [tex]\( 35^\circ \)[/tex].
- The supplementary angle is [tex]\( 145^\circ \)[/tex].
- The complementary angle is [tex]\( 55^\circ \)[/tex].
Therefore, the measure of the angle sought is [tex]\( 35^\circ \)[/tex].
