Question

two angles of a quadrilateral measure 120° and 160°. the other two angles are in a ratio of 3:17. what are the measures of those two angles?
□° and □°
submit

Explanation:

Step1: Recall quadrilateral angle sum

The sum of interior angles of a quadrilateral is \(360^\circ\).

Step2: Calculate sum of known angles

Sum of given angles: \(120^\circ + 160^\circ = 280^\circ\).

Step3: Find sum of unknown angles

Let the unknown angles be \(3x\) and \(17x\). Their sum is \(360^\circ - 280^\circ = 80^\circ\). So, \(3x + 17x = 80^\circ\).

Step4: Solve for x

Simplify: \(20x = 80^\circ\), so \(x = \frac{80^\circ}{20} = 4^\circ\).

Step5: Find the angles

First unknown angle: \(3x = 3\times4^\circ = 12^\circ\). Second unknown angle: \(17x = 17\times4^\circ = 68^\circ\). Wait, no, wait—wait, \(3x + 17x = 20x = 80^\circ\), so \(x = 4^\circ\)? Wait, no, that can't be. Wait, no, \(360 - 280 = 80\), so \(3x + 17x = 20x = 80\), so \(x = 4\). Then \(3x = 12\), \(17x = 68\)? Wait, no, that seems too small. Wait, no, wait, quadrilateral angle sum is \(360\). Wait, \(120 + 160 = 280\), so remaining is \(80\). So \(3x + 17x = 20x = 80\), so \(x = 4\). So angles are \(12^\circ\) and \(68^\circ\)? Wait, no, that can't be right. Wait, no, maybe I made a mistake. Wait, no, let's check again. \(3x + 17x = 20x\). \(20x = 80\), so \(x = 4\). So \(3x = 12\), \(17x = 68\). Wait, but \(12 + 68 = 80\), and \(120 + 160 + 12 + 68 = 360\). Yes, that works. Wait, but maybe I messed up the ratio. Wait, the ratio is 3:17, so 3 parts and 17 parts. So total parts 20. So 20 parts = 80 degrees, so 1 part = 4 degrees. So 3 parts = 12, 17 parts = 68. Yes, that's correct.

Answer:

\(12^\circ\) and \(68^\circ\)