**Question**

**Introduction**

This looks
challenging because we do not seem to be given much information. For example, we do not know the individual
angles of the

*trapezium*(American:*trapezoid*). Does this mean that this puzzle cannot be solved? Are we trapped by this trapezium? What is the secret key that unlocks the problem?**Solution**

First sketch trapezium mid-point
of . All contructed lines and points are shown in
grey (

*ABCD*. Introduce point*F*, the*AD*

*American:*gray). Then*DF*=*FA*=*BC*.
Now draw a line parallel to

*BC*passing through*A*and intersecting*DC*at*E*, say. Note that Ð*AED*= Ð*BCD*(corresponding angles). Then Ð*ADE*+ Ð*AED*= Ð*ADC*+ Ð*BCD*= 120°. Hence the remaining angle in D*DAE*, Ð*DAE*= 180° – 120° = 60°. We can deduce this, even though we do not (currently) know the individual angles Ð*ADE*and Ð*AED*. This is the key step.
From here, things get easier. D

*AFE*is an isosceles triangle with Ð*AFE*= Ð*AEF*= (180° – 60°) ¸ 2 = 60°. So D*AFE*is in fact an equilateral triangle. That means*FE*=*FA*=*FD*. So D*FDE*is an isosceles triangle.
Note that
Ð

*AFE*is an exterior angle of the triangle D*FDE*. If you know that exterior angle of a triangle is the sum of the interior opposite angles, then from Ð*AFE*= Ð*FED*+ Ð*FDE*, we easily see that Ð*ADE*= Ð*FDE*= 60° ¸ 2 = 30°. If you do not know the theorem about exterior angles, you can still quickly work out that Ð*DFE*= 120°, and then use Ð*ADE*= (180° – 120°) ¸ 2 and arrive at the same conclusion. ©**Remarks**

We can now see that actually Ð

*BCD*= Ð*AED*= 90°, but we do not need to rely on that (or on accurate drawing) to deduce the answer. From solving this question, we learn that even though we do not know the individual angles, by construction and using the sum of angles in a triangle, it is possible to solve for an important angle, namely Ð*AFE*. The rest of the solving uses isosceles triangles and equilateral triangles, which are part of the common repertoire of tactics.
H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in
another way

H10. Simplify the problem

H11. Solve part of the problem

H12* Think of a related problem (Draw construction lines)

**Suitable Levels**

*****Primary School Mathematics

* other syllabuses that involve area of triangles and circles

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