Answer: The first and second angles measure 54 degrees, and the third angle measures 72 degrees.
Step-by-step explanation: Given that two angles of a triangle have equal measures, but the third angle's measure is 36° less than the sum of the other two.
We are to find the measure of each angle of the triangle.
Let, x° be the measure of each of the two angles that has equal measure.
Then, the measure of the third angle will be (x° + x° - 36°) = (2x° - 36°).
From Angle-Sum-Property of a triangle, we have
![x^\circ+x^\circ+(2x^\circ-36^\circ)=180^\circ\\\\\Rightarrow 2x^\circ+2x^\circ-36^\circ=180^\circ\\\\\Rightarrow 4x^\circ=180^\circ+36^\circ\\\\\Rightarrow 4x^\circ=216^\circ\\\\\Rightarrow x=54^\circ.](https://tex.z-dn.net/?f=x%5E%5Ccirc%2Bx%5E%5Ccirc%2B%282x%5E%5Ccirc-36%5E%5Ccirc%29%3D180%5E%5Ccirc%5C%5C%5C%5C%5CRightarrow%202x%5E%5Ccirc%2B2x%5E%5Ccirc-36%5E%5Ccirc%3D180%5E%5Ccirc%5C%5C%5C%5C%5CRightarrow%204x%5E%5Ccirc%3D180%5E%5Ccirc%2B36%5E%5Ccirc%5C%5C%5C%5C%5CRightarrow%204x%5E%5Ccirc%3D216%5E%5Ccirc%5C%5C%5C%5C%5CRightarrow%20x%3D54%5E%5Ccirc.)
So, the measure of each angle of equal measure is 54°, and the measure of the third angle is
![2\times 54^\circ-36^\circ=108^\circ-36^\circ=72^\circ.](https://tex.z-dn.net/?f=2%5Ctimes%2054%5E%5Ccirc-36%5E%5Ccirc%3D108%5E%5Ccirc-36%5E%5Ccirc%3D72%5E%5Ccirc.)
Thus, the first and second angles measure 54 degrees, and the third angle measures 72 degrees.