Let's start by understanding the given information and breaking it down step-by-step.
We know that the sum of two angles, let's call them [tex]\( D \)[/tex] and [tex]\( G \)[/tex], is given as 80 degrees:
[tex]\[ D + G = 80 \][/tex]
We are also given a relationship between [tex]\( D \)[/tex] and [tex]\( G \)[/tex]:
[tex]\[ G = D + \frac{D}{9} \][/tex]
We need to prove this relationship and show how it holds.
### Step-by-Step Proof:
1. Express [tex]\( G \)[/tex] in terms of [tex]\( D \)[/tex]:
According to the given relationship:
[tex]\[ G = D + \frac{D}{9} \][/tex]
2. Substitute [tex]\( G \)[/tex]'s expression into the sum equation:
Since [tex]\( D + G = 80 \)[/tex], we substitute the expression for [tex]\( G \)[/tex]:
[tex]\[ D + \left( D + \frac{D}{9} \right) = 80 \][/tex]
3. Combine like terms:
Simplify the left side of the equation:
[tex]\[ D + D + \frac{D}{9} = 80 \][/tex]
[tex]\[ 2D + \frac{D}{9} = 80 \][/tex]
4. Make a common denominator:
To combine [tex]\( 2D \)[/tex] and [tex]\( \frac{D}{9} \)[/tex], find a common denominator, which is 9:
[tex]\[ 2D = \frac{18D}{9} \][/tex]
[tex]\[ \frac{18D}{9} + \frac{D}{9} = 80 \][/tex]
5. Combine the fractions:
Add the fractions together:
[tex]\[ \frac{18D + D}{9} = 80 \][/tex]
[tex]\[ \frac{19D}{9} = 80 \][/tex]
6. Solve for [tex]\( D \)[/tex]:
To find [tex]\( D \)[/tex], multiply both sides of the equation by 9:
[tex]\[ 19D = 720 \][/tex]
Divide both sides by 19:
[tex]\[ D = \frac{720}{19} \][/tex]
[tex]\[ D = 72 \][/tex] (since [tex]\( 720 \div 19 = 72 \)[/tex])
7. Find [tex]\( G \)[/tex] using [tex]\( D \)[/tex]:
Now that we have [tex]\( D = 72 \)[/tex]:
[tex]\[ G = D + \frac{D}{9} \][/tex]
Substitute [tex]\( D = 72 \)[/tex]:
[tex]\[ G = 72 + \frac{72}{9} \][/tex]
[tex]\[ G = 72 + 8 \][/tex]
[tex]\[ G = 80 \][/tex]
8. Verify the sum:
Check the sum [tex]\( D + G \)[/tex]:
[tex]\[ D + G = 72 + 80 = 80 \][/tex]
### Conclusion:
We have shown that, given [tex]\( D \)[/tex] and [tex]\( G \)[/tex] as the measures of the two angles with [tex]\( D + G = 80 \)[/tex] degrees, [tex]\( G \)[/tex] indeed equals [tex]\( D + \frac{D}{9} \)[/tex]. We computed [tex]\( D = 72 \)[/tex] degrees and [tex]\( G = 80 \)[/tex] degrees, which satisfies all conditions. This verifies the given relationship [tex]\( G = D + \frac{D}{9} \)[/tex] and shows how it holds true.
