A quadrilateral has two angles that measure 278° and 2°. The other two angles are in a ratio of 3:7. What are the measures of those two angles?
The sum of the angles in a quadrilateral is 360°.
Let's call the two unknown angles x and y.
We know that x:y is in a ratio of 3:7, so we can write x=3a and y=7a for some value of a.
Then we can set up an equation:
278° + 2° + x + y = 360°
Substituting in our expressions for x and y:
278° + 2° + 3a + 7a = 360°
Combining like terms:
280° + 10a = 360°
Subtracting 280° from both sides:
10a = 80°
Dividing by 10:
a = 8°
So now we know that x = 3a = 24° and y = 7a = 56°.
Therefore, the measures of the two unknown angles are 24° and 56°.
To find the measures of the other two angles, we can first subtract the two given angles from 360 degrees, since the sum of the angles in a quadrilateral is always 360 degrees.
So, subtracting 278° and 2° from 360°, we get: 360° - 278° - 2° = 80°.
Next, we find the ratio of the other two angles, which is 3:7. Since the total ratio is 3 + 7 = 10, we can divide 80° by 10 to find the value of one unit of the ratio.
80° ÷ 10 = 8°.
Therefore, the measures of the other two angles can be found by multiplying the unit value of the ratio (8°) by 3 and 7, respectively.
3 x 8° = 24°.
7 x 8° = 56°.
So, the measures of the other two angles are 24° and 56°.