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Critical Thinking Strategies: The Case for SAT Math

Today, Dr. Gruber, renowned SAT expert and pioneer of the critical thinking strategies and skills necessary for standardized tests, shares 2 key strategies for tackling a tough SAT math problem.

To begin, here’s an SAT problem that an enormous amount of students got wrong. In fact out of the 1,275,000 students who take the SAT, only a handful was able to answer it. Give it a try and see how you fare:


In the figure above, which is true?

  1. x + y + z=180-a
  2. 2x + y + z = a
  3. x – y + x – z = a
  4. x + y + z + a = 270
  5. x + y + z = a

Do you have your answer yet? It turns out that with just two powerful strategies, we can solve this classic problem.

The first strategy is to draw lines to complete a diagram and label parts. Draw line BC and label the extra angles, b and c. We get this image:


Next, use the known characteristic of any triangle: that the sum of the interior angles always adds up to 180°. We get

(1) x + y + b + z + c= 180 for the big triangle and

(2) a + b + c = 180 for the smaller triangle.

Now, use the second powerful strategy: Don’t just solve for variables especially when you have many of them. Just add or subtract equations.

In this case we would subtract equations to reduce the amount of variables.

When we subtract equation 2 from equation 1, we get:

x+y+b+z+c-a-b-c= 180 -180 = 0

We end up with: x + y + z –  a = 0


x + y + z = a   (choice E).

And that’s all it takes! Many of the strategies used to solve that problem can also be found in the video below, as well as in my books and website.


If you’d like to know more about Dr. Gary Gruber or if you’re interested in his individualized test preparation program, write him at garyg@drgarygruber.com. He takes only a limited amount of students!


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About Guest Author

This post was written by a friend of Magoosh.

2 Responses to “Critical Thinking Strategies: The Case for SAT Math”

  1. tina kolpakowski says:

    In this problem, you can substitute (360 – a) for the missing fourth angle in the quadrilateral, which gives you

    x + y + z + (360 – a) = 360.

    Simplifying gives you

    x + y + z = a

    No auxiliary lines needed!

  2. Jim Kosinski says:

    A very nice problem.
    I like the directness of Tina’s solution
    for those with good memories
    and the visual in yours.
    Here’s another visual approach based on symmetry;
    Draw a mirror image from the vertices of y and z
    to create a parallelogram. All the angles add up to 360.
    Students with a strength in art may find this an easy visual
    path to take.

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