Question

two angles of a quadrilateral measure 146° and 163°. the other two angles are in a ratio of 8:9. what are the measures of those two angles?
□° and □°

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: \(146^\circ + 163^\circ = 309^\circ\).

Step3: Find sum of unknown angles

Let the unknown angles be \(8x\) and \(9x\). Then \(8x + 9x + 309^\circ = 360^\circ\). Simplify: \(17x = 360^\circ - 309^\circ = 51^\circ\).

Step4: Solve for \(x\)

\(x=\frac{51^\circ}{17} = 3^\circ\).

Step5: Find unknown angles

First angle: \(8x = 8\times3^\circ = 24^\circ\). Second angle: \(9x = 9\times3^\circ = 27^\circ\). Wait, no, wait—wait, that can't be. Wait, no, I made a mistake. Wait, \(360 - 146 - 163 = 360 - 309 = 51\)? Wait, no, 146 + 163 is 309? 146 + 160 is 306, plus 3 is 309. Then 360 - 309 is 51? Wait, but 8x + 9x = 17x = 51? Then x = 3. So 8x is 24, 9x is 27? But that seems too small. Wait, no, wait, quadrilateral angles: maybe I miscalculated. Wait, 146 + 163: 146 + 160 = 306, +3 = 309. 360 - 309 = 51. Then 8x + 9x = 17x = 51, so x = 3. So 83=24, 93=27. Wait, but that seems odd. Wait, maybe I messed up the angle sum? No, quadrilateral is 360. Wait, maybe the problem is correct. Wait, but let's check again. 146 + 163 + 24 + 27 = 146+163=309, 24+27=51, 309+51=360. Yes, that works. Wait, but the angles seem small, but mathematically it's correct. Wait, no, wait, maybe I misread the ratio. Wait, the ratio is 8:9, so 8x and 9x. So 8x + 9x = 17x. 360 - 146 - 163 = 51. So x=3. So 24 and 27. Wait, but maybe the original problem had different numbers? Wait, no, the user's problem is as stated. So according to the calculation, the angles are 24° and 27°? Wait, no, that can't be. Wait, no, wait, 146 + 163 is 309, 360 - 309 is 51. 51 divided by 17 is 3. So 83=24, 93=27. Yes, that's correct.

Answer:

24° and 27°