The angle addition postulate in geometry states that if we place two or more angles side by side such that they share a common vertex and a common arm between each pair of angles, then the sum of those angles will be equal to the total sum of the resulting angle. For example, if ∠AOB and ∠BOC are adjacent angles on a common vertex O sharing OB as the common arm, then according to the angle addition postulate, we have ∠AOB + ∠BOC = ∠AOC.
The definition of angle addition postulate states that "If a ray is drawn from point O to point P which lies in the interior region of ∠MON, then ∠MOP + ∠NOP = ∠MON". This postulate can be applied to any pair of adjacent angles in math. In other words, the angle addition postulate can be defined as 'the sum of two angles joined together through a common arm and a common vertex is equal to the sum of the resulting angle formed'.
If an angle AOC is given where O is the vertex joining rays OA and OC, and there lies a point B in the interior of ∠AOC, then the angle addition postulate formula is given as ∠AOB+∠BOC = ∠AOC. If ∠AOC is divided into more than two angles such as ∠AOB, ∠BOD, and ∠DOC, then also we can apply the formula of angle addition postulate as ∠AOB+∠BOD+∠DOC = ∠AOC.
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Example 1: In the figure given below, if ∠POS is a right angle, ∠2 = 30°, and ∠3 = 40°. Find the value of ∠1. Solution: It is given that ∠POS is a right angle. It means that ∠POS = 90°. Now, by using the angle addition postulate formula, we can write ∠1 + ∠2 + ∠3 = 90°. Given, ∠2 = 30° and ∠3 = 40°. Substituting these values in the above equation, we get, ∠1 + 30° + 40° = 90° ∠1 + 70° = 90° ∠1 = 90° - 70° ∠1 = 20° Therefore, the value of ∠1 is 20°.
Example 2: In the given figure, XYZ is a straight line. Find the value of x using the angle addition postulate. Solution: It is given that XYZ is a straight line. It means that ∠XYO and ∠OYZ form a linear pair of angles. ⇒ ∠XYO + ∠OYZ = 180° (using angle addition postulate and linear pair of angles property) ⇒ (3x + 5) + (2x - 5) = 180° ⇒ 5x = 180° ⇒ x = 36 Therefore, the value of x is 36.
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