Question

describe the error in finding ( mangle1 ).
( 115^{circ}+39^{circ}+mangle1 = 360^{circ} )
( 154^{circ}+mangle1 = 360^{circ} )
( mangle1 = 206^{circ} )
the sum of ( 360^{circ} ) and ( 154^{circ} ) should be ( 514^{circ} ).
the sum of the measures of the angles should be ( 180^{circ} )
the sum of ( 115^{circ} ) and ( 39^{circ} ) should be ( 144^{circ} ).
the sum of the measures of the angles should be ( 90^{circ} ).

Explanation:

Step1: Recall the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). In the given problem, the equation \(115^{\circ}+39^{\circ}+m\angle1 = 360^{\circ}\) was used. This is incorrect because \(360^{\circ}\) is the sum of angles in a full - circle or the sum of exterior angles of a polygon (not relevant here). The correct sum for the interior angles of a triangle is \(180^{\circ}\).

Step2: Set up the correct equation

Using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), where \(A = 115^{\circ}\), \(B = 39^{\circ}\), and \(C=m\angle1\). The correct equation is \(115^{\circ}+39^{\circ}+m\angle1 = 180^{\circ}\).

Step3: Solve the correct equation

First, add \(115^{\circ}\) and \(39^{\circ}\): \(115^{\circ}+39^{\circ}=154^{\circ}\). Then, the equation becomes \(154^{\circ}+m\angle1 = 180^{\circ}\). Subtract \(154^{\circ}\) from both sides: \(m\angle1=180^{\circ}-154^{\circ}\). So, \(m\angle1 = 26^{\circ}\).

Answer:

The sum of the measures of the angles should be \(180^{\circ}\).