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These shapes have four sides and 360° of interior angle goodness.

In the above diagram, quadrilateral \(ABCD\) 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 \(\triangle OBC ?\)

**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."

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

**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."

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In the above diagram, quadrilateral \(ABCD\) 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 \(ABCD ?\)

**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."

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In the above diagram, quadrilateral \(ABCD\) 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 \(x+y ?\)

**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."

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Trapezium \(ABCD\) has \(AB\) parallel to \(DC\). If \(\angle DBC = 23 ^\circ\) and \(\angle BCD = 59 ^\circ\), what is \(\angle ABD\) (in degrees)?

**Details and assumptions**

A trapezium has a pair of parallel sides.

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