Sure, let’s solve the problem step-by-step.
Given:
1. The sum of two angles of a triangle is [tex]\(72^\circ\)[/tex].
2. The difference between these two angles is [tex]\(20^\circ\)[/tex].
Let’s denote the two angles as [tex]\( x \)[/tex] and [tex]\( y \)[/tex], where [tex]\( x \)[/tex] is the larger angle and [tex]\( y \)[/tex] is the smaller angle.
We can set up the following system of equations based on the problem statement:
1. [tex]\( x + y = 72 \)[/tex]
2. [tex]\( x - y = 20 \)[/tex]
To solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we will use these equations. First, add the two equations together:
[tex]\[ (x + y) + (x - y) = 72 + 20 \][/tex]
This simplifies to:
[tex]\[ 2x = 92 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{92}{2} = 46 \][/tex]
Next, we will substitute the value of [tex]\( x \)[/tex] back into the first equation to solve for [tex]\( y \)[/tex]:
[tex]\[ 46 + y = 72 \][/tex]
So:
[tex]\[ y = 72 - 46 \][/tex]
Thus:
[tex]\[ y = 26 \][/tex]
Therefore, the measures of the two angles are:
[tex]\[ x = 46^\circ \][/tex]
[tex]\[ y = 26^\circ \][/tex]
So, the two angles are [tex]\( 46^\circ \)[/tex] and [tex]\( 26^\circ \)[/tex].
