Question

Recall that two angles are complementary if the sum of their measures is 90degrees. Find the measures of two complementary angles if one angle is 6 degrees more than two times the other angle?

word problems

Answer 1

`28` and `62` degrees

Explanation:

Let `A=`the measure of the smaller angle
`B=`the measure of the larger angle

Since they are complementary `A+B=90`

We also know `2A+6=B`

Replace `B` in the first equation with the second equation:

`A+2A+6=90`

Solve for `A`:

`3A+6=90`
`3A=84`
`A=28`

So: `B=2(28)+6=62`

So the two angles are: `28` and `62` degrees

Answer 2

One is `28^o` and the other is `62^o`

Explanation:

Condition 1: Their sum is `90^o`

Dropping the degrees sign for now.

Condition 2:

Let the first angle be `x`
Let the second angle be `y`

Then form the question

"if one angle "`...................->x`
"is" `..................................->x=?`
"`6^o` more than "`.................->x=?+6`
" 2 times the other angle "`..->x=2?+6" "->" "x=2y+6`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This gives us two equations

`x+y=90" ".......................Equation(1)`
`x=2y+6" ".......................Equation(2)`

Method: change these so that we have 1 equation with 1 unknown and thus solvable.

`color(blue)("Determine the value of "y)`

Using `Equation(1)` to remove one of the unknowns in `Equation(2)`

Consider `Equation(1)`

Subtract `color(red)(y)` from both sides

#color(green)(x+y=90 color(white)("ddd")->color(white)("ddd") x+ycolor(red)(-y)=90color(red)(-y))#

`color(white)("dddddddddddd")->color(white)("dddd")x=90-y" ".....Equation(1_a)`

`color(brown)("Notice that this has the same consequence as the shortcut")` `color(brown)("approach of move to the other side and reverse the sign from + to -")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Using `Equation(1_a)` substitute for `x` in `Equation(2)`

`color(green)(color(red)(x)=2y+6color(white)("ddd")->color(white)("ddd")color(red)(90-y)=2y+6)`

Using the shortcut approach we have:

`84=3y`

The 3's sign is multiply so the shortcut is move to the other side and reverse the sign from `xx" to "-:` giving:

`84/3=y`

Write as `y=84/3` as this complies with convention

`color(white)("ddddddddddd")ul(bar(|color(white)("d")y=84/3 = 28color(white)("d")|))`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the value of "x)`

Using `Equation(1)`

`x+y=90`

`x=90-28`

`color(white)("ddddddddddd")ul(bar(|color(white)(2/2)x=62color(white)(2/2)|))`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Check")`

`2(color(red)(28))+6->56+6=62`