To find the supplement of [tex]\( 100^\circ 48' \)[/tex], let’s break the problem down into a few clear steps:
1. Convert degrees and minutes to decimal degrees:
- Degrees ([tex]\(d\)[/tex]): [tex]\( 100^\circ \)[/tex]
- Minutes ([tex]\(m\)[/tex]): [tex]\( 48' \)[/tex]
We know that 1 degree is equal to 60 minutes. So, to convert [tex]\( 48' \)[/tex] into degrees, we do:
[tex]\[
48' = \frac{48}{60} = 0.8^\circ
\][/tex]
Therefore, the total angle in decimal degrees is:
[tex]\[
100^\circ + 0.8^\circ = 100.8^\circ
\][/tex]
2. Calculate the supplement of the angle:
- The supplement of an angle is found by subtracting the angle from [tex]\( 180^\circ \)[/tex]:
[tex]\[
180^\circ - 100.8^\circ = 79.2^\circ
\][/tex]
3. Convert the supplement from decimal degrees back to degrees and minutes:
- Begin with the whole number part: [tex]\( 79 \)[/tex] degrees.
- The remaining decimal part needs to be converted back into minutes. The decimal part is [tex]\( 0.2 \)[/tex]:
[tex]\[
0.2^\circ \times 60 = 12 \text{ minutes}
\][/tex]
Therefore, the supplement of [tex]\( 100^\circ 48' \)[/tex] is [tex]\( 79^\circ 12' \)[/tex].
To summarize the results:
- Original angle: [tex]\( 100.8^\circ \)[/tex]
- The supplement in decimal degrees: [tex]\( 79.2^\circ \)[/tex]
- The supplement in degrees and minutes: [tex]\( 79^\circ 12' \)[/tex].
The complete answer is [tex]\( 79^\circ 12' \)[/tex] for the supplement of [tex]\( 100^\circ 48' \)[/tex].
