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Geometry

# Properties of Trapezoids (US) / Trapeziums (UK)

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."

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."

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."

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."

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

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