In triangle ABC, AC = BC, angle A = (x+24)/5 degrees, and angle B = (x-6)/2 degrees. How large is each angle?

In triangle ABC, Angle A is 10 degrees, Angle B is 10 degrees, and Angle C is 160 degrees. This is derived from the isosceles triangle property and solving the equations for the angles. By substituting x = 26 into the equations, we find the measures of all three angles.
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Answered by Anonymous | 2025-07-04

To solve for the angles in triangle A BC where A C = BC , we can use properties of isosceles triangles and the triangle angle sum theorem.

Isosceles Triangle Property : Since A C = BC , triangle A BC is isosceles, and angles A and B opposite these equal sides are equal. Thus, ∠ A = ∠ B .

Given Angle Expressions :



∠ A = 5 ( x + 24 ) ​

∠ B = 2 ( x − 6 ) ​



Equating the Angles : Set ∠ A equal to ∠ B .
5 ( x + 24 ) ​ = 2 ( x − 6 ) ​

Solving for x :



To clear the fractions, multiply through by 10 (the least common multiple of 5 and 2): 2 ( x + 24 ) = 5 ( x − 6 )

Simplify and solve the equation: 2 x + 48 = 5 x − 30 48 + 30 = 5 x − 2 x 78 = 3 x x = 26



Calculating the Angles :


Substitute x = 26 back into the expressions for ∠ A and ∠ B :

∠ A = 5 ( 26 + 24 ) ​ = 5 50 ​ = 10 degrees

∠ B = 2 ( 26 − 6 ) ​ = 2 20 ​ = 10 degrees



Finding ∠ C :


Use the triangle angle sum theorem, which states the sum of all angles in a triangle is 180 degrees. ∠ C = 18 0 ∘ − ∠ A − ∠ B = 18 0 ∘ − 1 0 ∘ − 1 0 ∘ = 16 0 ∘


Conclusion : The angles of triangle A BC are ∠ A = 10 degrees, ∠ B = 10 degrees, and ∠ C = 160 degrees.

Answered by danjohnbrain | 2025-07-06