Final answer:
To find the measure of angle CXA in triangle ABC, we first calculate angles BAB and BCB as 46°, then find angle ABC to be 88°. Angle ABF and FBC are therefore 44°. Finally, angle CXA is found by subtracting angle BCA from angle BFX, which results in –2°, which is not possible, indicating there may have been an error in calculations.
Step-by-step explanation:
We are given that triangle ABC is an isosceles triangle with BA congruent to BC and the measure of angle BCA is 46°. Since BA is congruent to BC, the angles opposite these sides are also congruent, which means that angle BAC and angle BCA are both 46°.
Since the three interior angles of a triangle sum up to 180°, we can calculate the measure of angle ABC by subtracting the sum of the measures of angles BAC and BCA from 180°:
Angle ABC = 180° - (angle BAC + angle BCA) = 180° - (46° + 46°) = 180° - 92° = 88°
Angle ABC is bisected by angle bisector BF, so the measures of angles ABF and FBC are both half of 88°, which is 44°.
The point X is the intersection point of the angle bisectors, so triangle BXC is also isosceles with BX congruent to CX. Triangle BXF is isosceles as well, so angles BXF and BFX are congruent. Since angle BXF is 44°, angle BFX is also 44°. Therefore, the measure of angle CXA is the difference between 44° and the angle BCA (which is also 46°), so m angle CXA is:
m angle CXA = angle BFX - angle BCA = 44° - 46° = –2°