Adjacent Angles Setup

Let the smaller angle be $x$ and the larger angle be $y$. The problem states two conditions:

• The sum of the two adjacent angles is $100^{\circ}$: $x + y = 100^{\circ}$.
• The larger angle ($y$) is $20^{\circ}$ less than five times the smaller angle ($x$): $y = 5x - 20^{\circ}$.

Solving for the Angles

We can solve this system of two equations. Substitute the expression for $y$ from the second equation into the first equation:

$x + (5x - 20^{\circ}) = 100^{\circ}$

Combine like terms:

$6x - 20^{\circ} = 100^{\circ}$

Add $20^{\circ}$ to both sides:

$6x = 120^{\circ}$

Divide by 6 to find the smaller angle:

$x = \frac{120^{\circ}}{6} = 20^{\circ}$

Calculating the Larger Angle

Now substitute the value of $x$ back into the equation for $y$:

$y = 5x - 20^{\circ}$

$y = 5(20^{\circ}) - 20^{\circ}$

$y = 100^{\circ} - 20^{\circ}$

$y = 80^{\circ}$

The larger angle is $80^{\circ}$.

Verification

Check if the conditions are met:

• Sum: $20^{\circ} + 80^{\circ} = 100^{\circ}$ (Correct)
• Relationship: $5(20^{\circ}) - 20^{\circ} = 100^{\circ} - 20^{\circ} = 80^{\circ}$ (Correct)

The larger angle is $80^{\circ}$.