Answer:
(b) Complementary; x = 5
Step-by-step explanation:
Angles are complementary when the total of their measures is 90°. Angles are supplementary when the total of their measures is 180°. A right angle has a measure of 90°, so any pair of angles that has the same total measure as a right angle are complementary angles.
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interpreting the figure
The little square in the corner of the figure indicates that angle is a right angle, and has a measure of 90°. The ray headed off to the northeast divides that 90° angle into two parts, labeled 65° and 5x°. In math, the whole is the sum of the parts, so the 90° angle is the sum of the 65° and 5x° angles.
Those two angles total 90°, so they are complementary.
making use of the relation
Now that you understand the sum of the two marked angles is 90°, you can use that fact to write an equation:
65° +5x° = 90° . . . . equation relating the complementary angle measures
solution
This is a 2-step linear equation, so is solved the way that kind of equation is solved.
5x° = 25° . . . . . step 1: add the opposite of 65° to get the x-term alone
x = 5 . . . . . . . . .step 2: divide by the coefficient of x to get x alone
The angles are complementary, and the value of x is 5.
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Additional comment
In order to keep the equation true, whatever operation we perform on one side of the equation must also be performed on the other side. When we say "add -65°", we mean that value is added to both sides of the equation. Above, we have skipped the steps of showing that addition and the subsequent simplification. (Similarly, we skipped the steps showing the division and the simplification process.)
65° +5x° = 90° . . the original equation. We want to eliminate the 65° term.
65° +(-65°) +5x° = 90° +(-65°) . . . . . add -65° (to both sides)
(65° -65°) +5x° = (90° -65°) . . . . . simplify the expressions a bit
0° +5x° = 25° . . . . . . . . . . . do the subtraction
5x° = 25° . . . . . . . . . . . 0 is the identity element for addition. Adding it changes nothing.
