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Heres an EASY-level Plane Geometry problem that could appear early on the ACT Math test.
Take one minute to try to work it and then I'll go through the solution.
The correct answer is C. This triangle has an interior angle that measures
60°, and an exterior angle that measures 95°. We're looking for the measure of the
angle x. Notice that these values are already shown on the drawing.
We know that the three angles in a triangle add up to 180°, but we only know one angle
at this point, so we can't use this yet.
The 95° exterior angle and one of the angles of the triangle, call it y, form a straight
line. We can't actually just assume they form a line by looking at the drawing; we have
to use the fact that the 95° angle is an EXTERIOR angle, which was stated in the problem.
An exterior angle is defined as the angle between an extended side of a triangle and
another side. This means exterior angles are always supplementary to an interior angle.
In this case, the 95° angle is supplementary to the one we labeled y, so these two angles
must add up to 180°. This gives us the equation y plus 95 is equal
to 180, OR y is equal to 180 minus 95, equals 85. y is 85 degrees.
Now that we know the measure of the angle y, we only have one unknown angle in the triangle,
x, the one we're looking for. So now's the time to use the fact that the angles in a
triangle sum to 180°. Adding the three angles gives 60 plus 85 plus
x, and that's got to be equal to 180 degrees. Subtracting 60 and 85 from both sides gives
us our answer, x=35 degrees
One caution about this problem: Notice that this triangle looks like a right triangle,
but the angle that appears to be about 90°, the one we labeled y, is really only 85°.
If you assumed that this was a right triangle just by looking at it, you would have gotten
an answer of x=30°, which is one of the wrong answer choices.
The ACT test writers intentionally include the most likely wrong answers in order to
catch people making common mistakes, like assuming that a triangle is a right triangle
just because it looks like one.
Two important geometry facts came up in this problem.
Supplementary angles are angles that form a line, they add to 180°. They can be in
any orientation, the thing to look for is that the two angles, taken together, form
a line. If they do, they add to 180°.
The second math fact that showed up in this problem is a basic property of triangles;
the three angles in a triangle add to 180°. This works for any triangle, including right
triangles.
Take a look at the little stretched out triangle on the far right. It has two very small angles,
and one really big one. The big angle looks like it measures a little
less that 180°. So the two small angles add just enough to
make the sum of the three angles exactly equal to 180°.
This always works: the angles in ANY triangle add up to 180°.