Two angles of a quadrilateral measure 140° and 50°. The other two angles are in a ratio of 7:10. What are the measures of those two angles?
Let the other two angles be 7x and 10x
The angles of a quad add up to 360°
so 140+50+7x+10x = 360
solve for x, then evaluate 7x and 10x
The interior angles of a quadrilateral measure to 360°
Two of them are known and are 140° and 50°
Sum of these two angles is:
140° + 50° = 190°
The other two angles must sum to:
360° - 190° = 170°
Mark the other two angles as A and B
A + B = 170°
The other two angles are in a ratio of 7:10 means:
A / B = 7 / 10
Cross multiply
10 A = 7 B
Divide both sides by 10
A = 7 / 10 B
Replace A with 7 / 10 B in formula
A + B = 170°
7 / 10 B + B = 170°
7 / 10 B + 10 B / 10 = 170°
17 B / 10 = 170°
Multiply both sides by 10
17 B = 1700°
Divide both sudes by 17
B = 100°
A = 7 / 10 B
A = 7 • 100° / 10
A = 700° / 10
A = 70°
The measures of other two angles are:
70° and 100°
To find the measures of the other two angles, we need to determine the ratio between them.
Let's denote the measures of the other two angles as x and y. According to the problem, x and y are in a ratio of 7:10. So we can write the equation:
x/y = 7/10
To solve this equation, we can cross-multiply:
10x = 7y
Now, we know that the sum of all the angles in a quadrilateral is 360 degrees. So we can write the equation:
140 + 50 + x + y = 360
Now, we can substitute the value of y from the first equation into the second equation:
140 + 50 + x + 10x/7 = 360
Simplifying the equation:
190 + (17x/7) = 360
To isolate x, we can subtract 190 from both sides:
(17x/7) = 170
Next, we can multiply both sides by 7 to cancel out the denominator:
17x = 1190
Finally, we can divide both sides by 17 to solve for x:
x = 1190 / 17 ≈ 70
Now, we can substitute the value of x back into the first equation:
70/y = 7/10
Cross-multiplying:
700 = 7y
Dividing both sides by 7:
y = 100
Therefore, the measures of the other two angles are approximately 70° and 100°.