Sum of Angles in a Triangle

Sum of Angles in a Triangle

Sum of Angles in a Triangle

The sum of all three angles in a triangle is always 180°. In the given figure ABC is a triangle. 
∠ABC + ∠BAC + ∠ACB = 180°

The sum of its all three angles is always 180° i.e. ∠ABC + BAC + ACB = 180°.

Theoretical proof:

Given: ABC is a triangle. BAC, ABC and ACB are interior angles of triangle ABC.
Triangle ABC
To prove: ABC + BAC + ACB = 180°
Construction: Through A, a line XAY parallel to BC is drawn.

Proof:
                Statements                                          Reasons
1.    XAB = ABC -------------------------> Alternate angles are equal.
2.    YAC = ACB -------------------------> Alternate angles are equal.
3.    XAB + BAC + YAC = 180° -----> Straight angle.
4.    ABC + BAC + ACB = 180° -----> From statements 1, 2 and 3.
                                                                                                            Proved.

Alternative method:

Given: ABC is a triangle. BAC, ABC and ACB are interior angles of triangle ABC.
Triangle ABC
To prove: ABC + BAC + ACB = 180°
Construction: A circle through A, B and C i.e. a circumcircle ABC is drawn.

Proof:
                Statements                                    Reasons
1.    ABC = ½ A͡C -------------------> Inscribed angle is half of opposite arc.
2.    BAC = ½ B͡C -------------------> Inscribed angle is half of opposite arc.
3.    ACB = ½ A͡C -------------------> Inscribed angle is half of opposite arc.
4.    ABC + BAC + ACB = ½ (A͡C + B͡C + A͡C) ---> Adding statements 1, 2 and 3.
                                                      = ½ ʘABC
                                                      = ½ × 360°
                                                      = 180°         Proved.                           
                                                                                                           

Workout Examples

Example 1: Find the unknown angle in the given figure.
Example 1: Triangle ABC
Solution: From the figure,
                          x + 50° + 70° = 180° ----------> Sum of angles in a triangles.
                or,     x + 120° = 180°
                or,     x = 180° – 120°
                or,     x = 60°


Example 2: Find the unknown angles in the given figure.
Example 2: Triangle ABC
Solution: From the figure,
                          x + 40° + 80° = 180° ----------> Sum of angles in a triangles.
                or,     x + 120° = 180°
                or,     x = 180° – 120°
                or,     x = 60°

                         y = 40° + 80° ------> exterior angle of a Δ is equal to the sum of opposite interior angles.
                or,     y = 120°

                x = 60° and y = 120°


Example 3: Find the values of x, y and z in the given figure.
Example 3: Triangle TUV
Solution: From the figure,
                          2x + 9° + x + x + 7° = 180° ----------> Sum of angles in a triangles.
                or,     4x + 16° = 180°
                or,     4x = 180° – 16°
                or,     4x = 164°
                or,     x = 164°/4
                or,     x = 41°

                          y = x + 7°------> Alternate angles.
        = 41° + 7°
        = 48°

                          z = 2x + 9°------> Alternate angles.
       = 2×41° + 9°
       = 82° + 9°
       = 91°

                 x = 41° and y = 48° and z = 91°


Example 4: Find the values of x and y in the given figure.
Example 4: Triangle PQR
Solution: From the figure,
                          x = y – 20° ----------> base angles of an isosceles Δ.

                          x + y – 20° + 120° = 180° ------> sum of angles of a Δ.
                or,     y – 20° + y – 20° + 120° = 180° ------> putting x = y – 20°.
                or,     2y + 80° = 180°
                or,     2y = 180° – 80°
                or,     2y = 100°
                or,     y = 100°/2
                or,     y = 50°

                          x = y – 20°
                             = 50° – 20°
                             = 30°

                x = 30° and y = 50°

Example 5: Find the values of a, and b in the given figure.
Solution: From the figure,
                          a + 50° + 60° = 180° ----------> Sum of angles in ΔABC.
                or,     a + 110° = 180°
                or,     a = 180° – 110°
                or,     a = 70°

                Again,
                          b + 60° + 30° = 180° ----------> Sum of angles in ΔADE.
                or,     b + 90° = 180°
                or,     b = 180° – 90°
                or,     b = 90°

                 a = 70° and b = 90°


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