Final answer:
The measure of angle CXA (m∠CXA) in an isosceles triangle ABC with angle bisectors intersecting at X is 68°. This is calculated by bisecting the known angles of the triangle and summing the resultant angles at point X.
Step-by-step explanation:
To find the measure of angle CXA (m∠CXA), we need to understand the properties of triangle bisectors and isosceles triangles. In an isosceles triangle, the angles opposite the equal sides are also equal. Since ABC is isosceles and angle bisectors α, β, and γ intersect at X, let's denote the measure of angle BCA as 44°.
Now, if we assume AB = AC, then angle ABC is also 44°. The angle bisector α cuts angle BAC into two equal parts. Since the sum of angles in any triangle is 180°, we can calculate the measure of angle BAC as follows: angle BAC = 180° - angle ABC - angle BCA = 180° - 44° - 44° = 92°.
As α bisects angle BAC, each half is 46°. Thus, the measure of angle BXA is also 46°. Since angle CXA is formed by the bisectors β covering angle ABC and α covering angle BAC, we can find its measure by adding the halves of angles ABC and BAC. Therefore, m∠CXA = 46° (half of BAC) + 22° (half of ABC) = 68°.