Geometry
# Quadrilaterals

$ABCD$ is a trapezoid with $\overline{AD} \parallel \overline{BC}.$ If $\lvert{\triangle ABC}\rvert=60, \lvert{\triangle OCD}\rvert=20,$ what is the area of $\triangle OBC ?$

In the above diagram, quadrilateral**Note $1$:** The above diagram is not drawn to scale, and $\lvert \triangle ABC \rvert$ denotes the area of $\triangle ABC.$

**Note $2$:** In the US a quadrilateral with one pair of parallel lines is called a "trapezoid." Elsewhere, a quadrilateral with one pair of parallel lines is called a "trapezium."

$ABCD$ is a trapezoid with $\overline{AB} \parallel \overline{CD}.$ If $\alpha^\circ={73}^\circ$ and $\gamma^\circ={135}^\circ,$ what is the value of $\omega-\beta ?$

In the above diagram, quadrilateral**Note:** In the US a quadrilateral with one pair of parallel lines is called a "trapezoid." Elsewhere, a quadrilateral with one pair of parallel lines is called a "trapezium."

$ABCD$ is a trapezoid with $\overline{AD} \parallel \overline{BC}.$ If $\lvert{\overline{AO}}\rvert:\lvert{\overline{OC}}\rvert=1:3, \lvert{\triangle OBC}\rvert=54,$ what is the area of quadrilateral $ABCD ?$

In the above diagram, quadrilateral**Note $1$:** The above diagram is not drawn to scale, and $\lvert{\triangle ABC}\rvert$ denotes the area of $\triangle ABC.$

**Note $2$:** In the US a quadrilateral with one pair of parallel lines is called a "trapezoid." Elsewhere, a quadrilateral with one pair of parallel lines is called a "trapezium."

$ABCD$ is an isosceles trapezoid with $\overline{AD} \parallel \overline{BC}.$ If $l=7, \alpha^\circ={25}^\circ, \beta^\circ={115}^\circ,$ what is the value of $x+y ?$

In the above diagram, quadrilateral**Note $1$:** The above diagram is not drawn to scale.

**Note $2$:** In the US a quadrilateral with one pair of parallel lines is called a "trapezoid." Elsewhere, a quadrilateral with one pair of parallel lines is called a "trapezium."