MHB Difference of angles in a triangle

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In triangle ABC, with angle CAB measuring 96 degrees, the angles BDE and CED are given as 24 and 18 degrees, respectively. To find the difference between the two smallest angles of the triangle, one can use the properties of angle bisectors and the fact that the sum of angles in a triangle is 180 degrees. By calculating the angles in triangle DEI, the missing angle EID is determined to be 138 degrees. The discussion emphasizes the need for a method to solve the problem without relying on Geogebra, suggesting the use of the angle bisector theorem for further calculations. Understanding these relationships is crucial for solving the triangle's angle measures effectively.
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In a triangle ABC, let D and E be the intersections of the bisectors of the angles ABC and ACB with the sides AC and AB, respectively. Knowing that the measures in degrees of the angles BDE and CED are equal to 24 and 18, respectively, calculate the difference in degrees between the measures of the two smallest angles of the triangle. CAB angle = 96 degrees.

I checked with Geogebra:
Geogebra online(7).png

But how to solve this problem without Geogebra?
 
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Recalling that plane Euclidean triangle angles add up to 180 degrees, opposite angles in intersecting lines are equal and angle bisectors split angles in half, we can use inspection to determine all the angles in the diagram.

As an example, by inspection, triangle DEI has two angles shown so we can compute the missing angle EID as 180-24-18 = 138 degrees.
 
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so how to calculate for example angle ABC?
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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