Question

prove that the sum of all angles in triangle is 180​

Answer 1

Answer:

Step-by-step explanation:

Draw a line parallel to side BC of the triangle that passes through the vertex A. Label the line PQ. Construct this line parallel to the bottom of the triangle.

angle PAB + angle BAC + angle CAQ = 180 degrees.

Remember, all of the angles that comprise a straight line must be equal to 180°. Because angle PAB, angle BAC, and angle CAQ combine together to make line PQ, their angles must sum to 180°. Call this Equation

angle PAB = angle ABC and angle CAQ = angle ACB. Because you constructed line PQ parallel to side BC of the triangle, the alternate interior angles (PAB and ABC) made by the transversal line (line AB) are congruent. Similarly, the alternate interior angles (CAQ and ACB) made by the transversal line AC are also congruent

↑Equation 2: angle PAB = angle ABC

↑Equation 3: angle CAQ = angle ACB

↑It is a geometric theorem that alternate interior angles of parallel lines are congruent.[4]

⇔Substitute angle PAB and angle CAQ in Equation 1 for angle ABC and angle ACB (as found in Equation 2 and Equation 3) respectively. Knowing that the alternate interior angles are equal lets you substitute the angles of the triangle for the angles of the line.

↑Thus we get, Angle ABC + angle BAC + angle ACB = 180°.

↑In other words, in the triangle ABC, angle B + angle A + angle C = 180°. Thus, the sum of all the angles of a triangle is 180°.