Final answer:
The measures of the two angles in a quadrilateral with angles of 290° and 20°, and the other two angles in a ratio of 2:3 are 52° and 78°.
Step-by-step explanation:
A quadrilateral with two angles measuring 290° and 20° has the other two angles in a ratio of 2:3. To find the measures of these two angles, we can first find the sum of the measures of the two known angles, which is 290° + 20° = 310°. The other two angles have a sum of 180° - 310° = -130°.
Since angles cannot have negative measures, we can take the absolute value of -130°, which gives us 130°. To find the measures of the individual angles, we can set up the ratio as 2x + 3x = 130°, where x represents the common ratio. Solving this equation, we get 5x = 130°, and x = 26°. Therefore, the two angles are 2x = 52° and 3x = 78°.