Let's solve this problem step-by-step by using algebra.
1. Define the Variables:
- Let [tex]\( x \)[/tex] represent the measure of the larger angle in degrees.
- Let [tex]\( y \)[/tex] represent the measure of the smaller angle in degrees.
2. Set Up the Equations:
- According to the problem, the sum of the two angles is [tex]\( 99^\circ \)[/tex]. This gives us the first equation:
[tex]\[
x + y = 99
\][/tex]
- The second piece of information is that the difference between the larger angle and the smaller angle is [tex]\( 18^\circ \)[/tex]. This gives us the second equation:
[tex]\[
x - y = 18
\][/tex]
3. Solve the System of Equations:
- We now have a system of two linear equations:
1. [tex]\( x + y = 99 \)[/tex]
2. [tex]\( x - y = 18 \)[/tex]
- Adding these two equations eliminates [tex]\( y \)[/tex]:
[tex]\[
(x + y) + (x - y) = 99 + 18
\][/tex]
[tex]\[
2x = 117
\][/tex]
[tex]\[
x = \frac{117}{2}
\][/tex]
[tex]\[
x = 58.5
\][/tex]
- Substitute [tex]\( x \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[
58.5 + y = 99
\][/tex]
[tex]\[
y = 99 - 58.5
\][/tex]
[tex]\[
y = 40.5
\][/tex]
4. Find the Third Angle:
- The sum of all angles in a triangle is [tex]\( 180^\circ \)[/tex]. Thus, the third angle [tex]\( z \)[/tex] is given by:
[tex]\[
z = 180 - x - y
\][/tex]
[tex]\[
z = 180 - 58.5 - 40.5
\][/tex]
[tex]\[
z = 180 - 99
\][/tex]
[tex]\[
z = 81
\][/tex]
5. Conclusion:
- Therefore, the measures of the three angles in the triangle are:
[tex]\[
\angle x = 58.5^\circ, \quad \angle y = 40.5^\circ, \quad \text{and} \quad \angle z = 81^\circ.
\][/tex]
These are the angles of the triangle in degrees.
