Problem number 14. Luis places three rectangular pictures in the way shown. What is the value of X? To solve this problem, we can notice that the sum of the angles along a straight line is equal to 180 degrees. Let's look at the corner of the picture of the clock next to the angle marked X. You will see that in the larger picture there is another smaller angle marked. What is the value of that angle? Well, we know that the sum of the angle along a straight line is 180 degrees. 90 of these degrees are already taken by the corner of the clock picture, so that means there are 90 degrees left. One of these is given to angle X, and the remainder has to be whatever is left over, and these two need to add together to make 90. That is, this angle must be 90 minus X degrees, because 90 minus X plus X will make 90, and if we add the corner 90 more degrees, then we will get 180. By the same logic, let's look at the other angle next to the 62 degree angle in the bottom left corner. Well, we know that that side should also add to 180 degrees, and 90 degrees are already taken up by the corner of the picture of the kangaroo and the beaver. So this means that the remaining angle has to add up with 62 to make a 90. That is, the missing angle is 90 minus 62, and 90 minus 62 is 28. So now we've figured out all the angles inside the center quadrilateral, which I've marked in red here. We know all the angles in a quadrilateral always sum to 360 degrees. This is a rule of geometry. So now we already know all of the angles in the quadrilateral, and we can add them up. The angles in the quadrilateral are 28, 42, 270, and 90 minus X. Note that the angle of 270 comes from the reflex angle of the corner of the picture of the ship. We know that that inside angle of the corner of the ship picture is 90 degrees, so the outside angle, which is the part inside the quadrilateral, will be 360 minus 90, which will be 270 degrees. So the angles of the quadrilateral, just to repeat, are 28, 42, 270, and 90 minus X, and we know those should add up to 360. Let's try and solve for X. First, let's combine the first two terms on the right side, 270 and 90. Those two terms add up to 360, but we'll notice that that 360 matches exactly with the 360 we already have on the left side, and so that means that these two terms, the 360 on the left before the minus X and the 360 on the right, will both cancel out. Now we just have on the left side 28 plus 42, and we can move the minus X over to the right side to get 28 plus 42 is equal to X. So now we can just add 28 and 42, which is 70, and that gives us the equation that X is 70. So the answer is B, 70.

geometric arrangement

rectangular pictures

angle sums

quadrilateral

calculation