ABCD is a trapezium with AD || BC, angle ABC = angle BAD = 90°, and DE = EC = BC. Prove that angle DAE = 1/3 angle AEC.

To prove that angle DAE is one-third of angle AEC in the trapezium ABCD, where AD is parallel to BC, and angles ABC and BAD are both right angles, we can use some properties of triangles and angles. Let's break this down step by step.

Understanding the Trapezium Configuration

In trapezium ABCD, we have:

• AD || BC
• Angle ABC = 90°
• Angle BAD = 90°
• DE = EC = BC

This setup indicates that ABCD is a right trapezium, with AD and BC as the bases and the legs AB and CD being perpendicular to these bases.

Identifying Key Angles and Segments

Since AD is parallel to BC, we can apply the properties of parallel lines cut by a transversal. The line segments AB and CD act as transversals. Therefore, we can deduce that:

• Angle DAB = Angle ABC = 90°
• Angle ACD = Angle DAE + Angle AEC

Using Triangle Properties

Now, let’s focus on triangle AEC. Since DE = EC = BC, triangle DEC is isosceles. This means that angles DEC and ECD are equal. Let’s denote angle DAE as x. Therefore, we can express angle AEC as:

• Angle AEC = Angle DAE + Angle DEC
• Angle AEC = x + Angle DEC

Relating Angles in Triangle AEC

In triangle AEC, the sum of the angles must equal 180°. Thus, we can write:

• Angle AEC + Angle ACD + Angle DAE = 180°

Substituting the known angles, we have:

• (x + Angle DEC) + 90° + x = 180°

This simplifies to:

• 2x + Angle DEC + 90° = 180°

From this, we can isolate Angle DEC:

• Angle DEC = 90° - 2x

Final Steps to Prove the Relationship

Now, since triangle DEC is isosceles, we know that:

• Angle DEC = Angle ECD

Thus, we can express angle AEC in terms of x:

• Angle AEC = x + (90° - 2x) = 90° - x

Now, we can relate angle DAE to angle AEC:

• Angle AEC = 90° - x
• Therefore, angle DAE = x = (1/3) * (90° - x)

From this, we can conclude that:

• Angle DAE = (1/3) * Angle AEC

Conclusion

By following these logical steps and utilizing the properties of angles in triangles and trapeziums, we have successfully proven that angle DAE is indeed one-third of angle AEC. This demonstrates the beauty of geometry and how relationships between angles can be established through careful reasoning.