Question 29. An isosceles triangle ABC with AB equal to AC is split into three smaller isosceles triangles as shown, so that AD equals DB, CE equals CD, and BE equals EC. Note that the diagram is not drawn to scale. What is the size in degrees of angle BAC? Start off by redrawing this triangle. It is important to note that it is an isosceles triangle made up of more isosceles triangles. Let's start off by labeling angle BAC as alpha. Since it is isosceles, angle ABD will also equal alpha, and one of the rules of a triangle is the three angles always total 180 degrees. So we know that this will be 180 minus 2 alpha. Another rule that is important to note here is that a straight line has an angle of 180, so we can name this 2 alpha. Next, let's call angle DBC beta, and since this is also an isosceles triangle, we can label angle ECB as beta as well, and then we can label this as 180 minus 2 beta, and following the same rule as before, label this as 2 beta. We know that this is also an isosceles triangle, so these two angles will be equal to each other, and this lets us know that beta is equal to alpha, like so. With this information, we can clear out some of these and get to figuring out what our answer will be. We know that alpha plus alpha plus alpha plus beta plus beta equals 180, the total of angles in a triangle. Since alpha equals beta, we can convert the betas into alphas, and we get alpha plus alpha plus alpha plus alpha plus alpha equal to 180, simplifies as 5 alpha equal to 180 degrees. We simply have to divide by 5, and that gives us our answer, which will be E, 36.

isosceles triangle

angle calculation

geometry

symmetry

angle BAC