SAT Geometry Student-Produced Response

Correct Answer: 30

Solution:

Refer to the diagram below.

Because the bigger triangle is isosceles, $2y+2x = 180$ and $y=\frac{180-2x}{2} = 90-x.$

Likewise, because the smaller triangle is isosceles, $4x + 2z = 180,$ or $z = \frac{180-4x}{2} = 90-2x.$

$\begin{aligned} 30+z &=y\\ 30+90-2x &= 90-x\\ 120 -2x &=90-x\\ 30 &=x\\ \end{aligned}$

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Common Mistakes:

• Not knowing the properties of isosceles triangles.
• Not accounting for the two equal angles in either or both isosceles triangles.
• Not realizing that $30+z=y.$

If you got this problem wrong, you should review SAT Triangles .

Here we can use a circle theorem to get straight to the answer. The centre of a circle lies at angle 4x and goes through all points in the triangle. Now denote the vertices of the larger triangle as ABC clockwise (i.e. A is the top vertex) and let O be the centre.

Now by constructing a circle and joining OA it is clear to see that: $radius = OA = OB = OC \Rightarrow\ \angle ABO = \ angle BAO$ Due to symmetry through OA: $x =\boxed{30}$

By symmetry the reflex angle of 4x is 300 – 2x. So 360 = 300 – 2x + 4x, x=30