Answer:
x = 11, y = 4
Explanation:
You want to find x and y given an inscribed quadrilateral with angles identified as L=(10x), M=(10x-6), N=(16y+6), X=(4+18y).
Inscribed angles
The key here is that an inscribed angle has half the measure of the arc it subtends. Translated to an inscribed quadrilateral, this has the effect of making opposite angles be supplementary.
This relation gives you two equations in x and y:
- (10x) +(16y +6) = 180
- (10x -6) +(4 +18y) = 180
Elimination
Subtracting the first equation from the second gives ...
(10x +18y -2) -(10x +16y +6) = (180) -(180)
2y -8 = 0
y = 4
Substitution
Using this value of y in the first equation, we have ...
10x +(16·4 +6) = 180
10x +70 = 180
x +7 = 18
x = 11
The solution is (x, y) = (11, 4).
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Additional comment
The angle measures are L = 110°, M = 104°, N = 70°, X = 76°.
The "supplementary angles" relation comes from the fact that the sum of arcs around a circle is 360°. Then the two angles that intercept the major and minor arcs of a circle will have a total measure that is half a circle, or 180°.
For example, angle L intercepts long arc MNX, and opposite angle N intercepts short arc MLX.
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