Triangles ACD and BCD are isosceles. Angle BAC has a measure of 18 degrees and angle BDC has a measure of 48 degrees. Find the measure of angle ABD.

Question

Triangle BCD is inscribed in triangle ACD. B is the center point.

Answer 1

Hi Kate H.,

Always, always draw your figure. Let A be at the top, B directly below it, and C and D be laterally symmetrically below B (one to each side). Right away you should be starting to use what you know. Assume BC = BD (as segment lengths).

Now why did I assume that? Because the problem requires isosceles, and the given information seems to be imposing a symmetry on the figure.

Then angle BDC = angle DCB by symmetry. Then you can calculate angle CBD (I sure hope. Triangle angles add to 180). Then angle ABC and angle ABD equally split the circle around point B, and you can figure them knowing angle CBD (circle totals 360).

Notice: you didn't even need angle BAC. Because you weren't asked to figure angle ACB. So if you had started with that given (18), rather than with the 48, you would have been at a dead end. That is why a drawn figure is so useful -- even if you drew it out-of-proportion, you could still easily jump around visually from the desired result, back to the correct given (angle and measure) to start with.