# Geometry Problem 1513: Solving the base in a Right Trapezoid with Double Angle and Sum of Two Sides. Difficulty Level: High School.

In a right trapezoid ABCD, where angles C and D measure 90 degrees and angle BAD is twice angle CBD, if the sum of AB and AD is 36 units, calculate the measure of BC in units.

## Definitions and Suggestions

- A triangle is a polygon that has three sides, three vertices, and three angles. Triangles are some of the most basic shapes in geometry and can be found in many different forms.
- A trapezoid, also known as a trapezium in
British English, is a quadrilateral with at least one pair of parallel
sides. The parallel sides are called the bases of the trapezoid, and the
other two sides are called the legs.
- A right trapezoid, also known as a right trapezium,
is a trapezoid with two adjacent right angles. In other words, one of its legs
is perpendicular to both of its bases.
- An isosceles triangle is a type of triangle that has two sides of equal length and two angles opposite those sides of equal measure.
- In an isosceles triangle, the external double angle
opposite to the base is the external angle at one of the vertices of the
triangle that is opposite to the base.
- Two triangles are said to be congruent if all
corresponding sides and angles of one triangle are equal to the
corresponding sides and angles of the other triangle. There are several ways
to prove that two triangles are congruent, including the Side-Side-Side
(SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Hypotenuse-Leg
(HL) criteria.

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Thematic Poem:
Solving the base in a Right Trapezoid with Double Angle and Sum of Two Sides

In a realm of shapes and lines

Geometry's secrets, it defines

A
right trapezoid, its sides known

But its base, a mystery, is shown

With a double angle to behold

And a sum of sides to be told

The
puzzle lies in its base

A solution we must embrace

Through calculation and deduction

We seek a geometric construction

Lines and angles we align

To unlock the answer, divine

With perseverance and wit

We solve the base, bit by bit

And
with newfound clarity and grace

We unveil geometry's hidden face

For in this world of shapes and math

Problems are but a simple
path

To unlock the secrets of the universe

And in our minds, a new
knowledge immerse.