Two sides of a triangle measure 4cm and 9cm. Determine which of the following choices cannot be the measure of the third angle
Let's call the two sides of the triangle with measures 4cm and 9cm side A and side B, respectively.
In any triangle, the sum of all three angles is 180 degrees.
Let's assume that the third side of the triangle has length x cm.
According to the triangle inequality theorem, the sum of any two sides of a triangle must be greater than the third side. So, we can write the following inequality:
4 + x > 9 ---> x > 5
Now, let's find the range of possible values for x to form a triangle:
9 + x > 4 ---> x > -5
Since length cannot be negative, we will disregard the negative solution and focus on the positive range of values: x > 5.
Using the sum of the three angles equation, we can find the measure of the third angle:
Third angle = 180 - angle A - angle B
Since angle A and angle B are acute angles (less than 90 degrees due to the triangle inequality theorem), the third angle must be greater than 0 degrees and less than 180 degrees.
Now, let's examine the answer choices to determine which one cannot be the measure of the third angle:
A) 90 degrees - Possible
B) 110 degrees - Possible
C) 150 degrees - Possible
D) 180 degrees - Not Possible
Therefore, the measure of the third angle in a triangle with sides measuring 4cm and 9cm cannot be 180 degrees.
D) 180 degrees is the choice that cannot be the measure of the third angle.