When the measures of the angles of a triangle are placed in order, the difference between the middle angle and smallest angle is equal to the difference between the middle angle and largest angle. If one of the angles of the triangle has measure 23 degrees, then what is the measure in degrees of the largest angle of the triangle?

Question

When the measures of the angles of a triangle are placed in order, the difference between the middle angle and smallest angle is equal to the difference between the middle angle and largest angle. If one of the angles of the triangle has measure 23 degrees, then what is the measure in degrees of the largest angle of the triangle?

Answer 1

clearly, 23 is not the largest. If it is the smallest, then

23 + 23+d + (23+d)+d = 180
3d + 69 = 180
3d = 111
d = 37

and the angles are 23,60,97

Answer 2

Let the middle angle have measure $x$, and let the difference between this measure and each of the others be $y$. So, the angle measures are $x-y, x, x+y$. These must sum to $180^\circ$ because they are the measures of the angles of a triangle. So, we must have

\[(x-y)+x + (x+y) = 180^\circ.\]Simplifying the left-hand side gives $3x = 180^\circ$, so $x = 60^\circ$ is the measure of the middle angle. Therefore, the $23^\circ$ is the smallest angle, and is $60^\circ - 23^\circ = 37^\circ$ less than the middle angle. So, the largest angle has measure $60^\circ + 37^\circ = \boxed{97^\circ}$.

Answer 3

To solve this problem, we need to use the information given to find the measure of the largest angle of the triangle. Let's start by setting up some equations.

Let's assume that the angles of the triangle are A, B, and C, where A is the smallest angle, B is the middle angle, and C is the largest angle.

According to the given information, the difference between the middle angle (B) and smallest angle (A) is equal to the difference between the middle angle (B) and largest angle (C). Mathematically, we can write this as:

B - A = B - C

Simplifying the equation, we get:

-A = -C

This implies that the smallest angle (A) is equal to the largest angle (C).

Now, we are given that one of the angles of the triangle has a measure of 23 degrees. Since we already established that the smallest angle (A) is equal to the largest angle (C), we can conclude that the measure of the largest angle (C) is also 23 degrees.

Therefore, the measure of the largest angle of the triangle is 23 degrees.