Question

In triangle ABC, the measure of angle ACB is 42°. The angle bisectors AD and BE intersect at point O so that AE + OE = AB. Find the measure of angle ABC and CAB.

a) ABC = 21°, CAB = 21°
b) ABC = 42°, CAB = 21°
c) ABC = 42°, CAB = 42°
d) ABC = 21°, CAB = 42°

Answer 1

The measure of angle CAB is 84° and angle ABC is 54°, which can be determined by applying properties of angle bisectors and isosceles triangles within the given triangle ABC.

Step-by-step explanation:

The student has asked to find the measure of angles ABC and CAB in a triangle ABC, given the measure of angle ACB is 42° and that the angle bisectors AD and BE intersect at a point O where AE + OE = AB.

Since angles AD and BE are angle bisectors, they divide angles CAB and CBA into equal parts respectively. Accordingly, let x be the measure of the angles CAD and DAB (since AD bisects angle CAB), and let y be the measure of the angles CBE and EBA (since BE bisects angle CBA). Because AE + OE = AB, and because angle bisectors in a triangle intersect at the incenter which equidistant from all sides of the triangle, it follows that triangle AOE is isosceles, and thus, angles EAO and EOA are equal. Since the sum of angles in triangle AOE must equal 180°, angles EAO and EOA each measure 42°, and therefore, angle CAB (x + x) is 84°. Now, knowing that the sum of angles in any triangle is 180°, we can solve for y:

180° = 42° + 2x + 2y
180° = 42° + 84° + 2y
180° - 126° = 2y
54° = 2y
y = 27°

Thus, angle CAB is 84° and angle ABC is 54°, making none of the given options correct.