Question

In a quadrilateral, two angles are equal. The third angle is the sum of the two

equal angles. The fourth angle is 60º less than twice the sum of the other

angles. Find the measure of the largest angle. Make a sketch.

Answer 1

The sum of the interior angles of a quadrilateral is 360°.

$$\alpha+\beta+\gamma+\theta=360\degree$$

Given

$$\alpha=\beta, \gamma=\alpha+\beta$$

$$\theta=2(\alpha+\beta+\gamma)-60\degree$$

$$\alpha+\beta+\gamma=360\degree-\theta$$

Substitute

$$\theta=2(360\degree-\theta)-60\degree$$

$$3\theta=660\degree$$

$$\theta=220\degree$$

$$\alpha+\beta+\gamma=360\degree-220\degree=140\degree$$

$$\alpha+\alpha+2\alpha=140\degree$$

$$\alpha=35\degree=\beta, \gamma=2(35\degree)=70\degree$$

The angles are

$$35\degree, 35\degree, 70\degree, 220\degree$$