SOLUTION: QUESTION: In isosceles triangle ABC the ratio of the measure of vertex angle A to the measure of angle B is 2 to 5. Find angle C. <br>1st APPROACH: we know that in an isosceles

Question 165224This question is from textbook Lets Review Math A
: QUESTION: In isosceles triangle ABC the ratio of the measure of vertex angle A to the measure of angle B is 2 to 5. Find angle C.

1st APPROACH: we know that in an isosceles triangle two angles are the same and the sides opposite them are also the same.

Since this problem doesn't tell you which is the base vertex in this isosceles triangle, I have assumed it to be 5x. Because of "RULE:"
the length of each side is less than the sum of the lengths of the other two sides and greater than the difference between these lengths....3less than x less than 7. Therefore x is in the range of 4, 5, 6.

2x+5x+5x=180

12x=180

x=15
Angle C is 75 degrees

2nd APPROACH: we know that in an isosceles triangle two angles are the same and the sides opposite them are also the same.

Since we already used 5x as the base vertex in our previous approach lets assume 2x is the base angle, this time, because it's an isosceles triangle.

2x+2x+5x=180

9x=180

x=20
Angle C is 40 degrees

BOOKS ANSWER IS 100 degrees, why?
This question is from textbook Lets Review Math A

Answer by jim_thompson5910(21667) (Show Source):
You can put this solution on YOUR website!
Here's a quick note about what the ratio is really saying:

Since the ratio is vertex angle A to the measure of angle B, this means that we have this ratio:

A:B ---> 2:5

So this means that the vertex angle is SMALLER than the base angle. However, if you have a base angle of 40 degrees, this means that the vertex angle is which means that the vertex angle is larger (not smaller) than the base angle. So this effectively rules out the "2nd Approach" answer. Remember, the ratio is vertex:base angle (not the other way around)

However, I agree with your first approach. That answer is correct. You'll find that the base angles are 75 degrees and the vertex angle is 30 degrees. Since the ratio of 30:75 reduces to 2:5, this verifies the answer.

Note: it is IMPOSSIBLE to have both an isosceles triangle AND have a base angle that is greater than 90 degrees. Why? Base angles are equal. So if there is an obtuse base angle, there are really 2 obtuse angles in the triangle (which is impossible; triangles can only have at most one obtuse angles). So this means that either the book is wrong or maybe you read the wrong answer.