The measures of the angles xyz are in the ratio 4 5 6 what are the measures of the angles

4 : 5 : 6
We know that the angles of the triangles add up to 180°

So do this ratio:
4a : 5a : 6a = 180°
4a + 5a + 6a = 180
15a = 180

Divide by 15:
= 
15 and 15 cancels out

a = 12

Plug in a:

4a : 5a : 6a
4(12) : 5(12) : 6(12)
48 : 60 : 72

These are the measure of the angles.
48°
60°
72°

check:
48 + 60 + 72 = 180
180 = 180

Wow. that is a really cool question.

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