# Two angles are complementary. The sum of the measure of the first angle and one-fourth the second angle is 69 degrees. What is the measures of the angles?

These are now 2 equations in 2 unknowns which may be solved simultaneously to get :

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Let the first angle be represented as ( x ) degrees and the second angle be represented as ( y ) degrees.

Given that the angles are complementary, we have the equation: [ x + y = 90 ]

Also, it is given that the sum of the measure of the first angle and one-fourth the second angle is 69 degrees, so we have: [ x + \frac{1}{4}y = 69 ]

To solve this system of equations, we can use substitution or elimination.

Substituting ( y = 90 - x ) from the first equation into the second equation: [ x + \frac{1}{4}(90 - x) = 69 ]

Solve for ( x ): [ x + 22.5 - \frac{1}{4}x = 69 ] [ \frac{3}{4}x = 46.5 ] [ x = 62 ]

Substitute ( x = 62 ) into the first equation to find ( y ): [ 62 + y = 90 ] [ y = 28 ]

So, the measures of the angles are ( 62^\circ ) and ( 28^\circ ).

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