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Adjacent angles

Adjacent angles. Adjacent angles. If two angles have the same vertex and a common ray, then the angles are called adjacent angles. In Fig, ∠BAC and ∠CAD are adjacent angles (i.e ∠ x and ∠y) as they have common ray. Adjacent angles on a line.

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Adjacent angles

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  1. Adjacent angles

  2. Adjacent angles If two angles have the same vertex and a common ray, then the angles are called adjacent angles. In Fig, ∠BAC and ∠CAD are adjacent angles (i.e ∠ x and ∠y) as they have common ray.

  3. Adjacent angles on a line When a ray stands on a straight line two angles are formed. They are called linear adjacent angles on the line. In Fig, ∠AOC and ∠BOC are adjacent angles on a line as they have common ray.

  4. The sum of the adjacent angles on a line is 180° ∠ AOB=1800 is a straight angle. In fig. The ray OC stands on a line AB. ∠AOC and ∠COB are adjacent angles. ∠AOB is a straight angle whose measure is 180° So, ∠AOC + ∠COB = 180° Thus the sum of the adjacent angles on a line is 180° Note 1: A pair of adjacent angles whose non common rays are opposite rays. Note 2: Two adjacent supplementary angles form a straight angle.

  5. Adjacent angles example Example 1: From the figure, Identify a) Two pairs of adjacent angles. • Solution: • Two pairs of Adjacent angles. • ∠EOC and ∠BOE (As shown in the fig, ∠1 and ∠2 are adjacent) • (since OE is common to ∠EOC and ∠ BOE) • ∠BOC and ∠BOD (As shown in the fig, ∠3 and ∠4 are adjacent) • (since OB is common to ∠BOC and ∠BOD)

  6. Linear Pair The adjacent angles that are supplementary lead us to a pair of angles that lie on straight line. This pair of angles is called linear pair. From the figure, we can see that ∠ACD = 123o and ∠BCD = 57o If we add ∠ACD and ∠BCD, we should get 180o ∠ACD + ∠BCD = 180o 123 + 57 = 180o Therefore, ∠ACD and ∠BCD are linear pairs D 123o 57o C A B

  7. Example 2: Check if the following pairs of angles form a linear pair i) 76, 129 ii) 30, 150 Solution: • 76, 129 Let us add 76 and 129 to see if we get 180 76 + 129 = 205 205 = 180 Hence, 76 and 129 are not a linear pair. ii) 30, 150 Let us add 30 and 150 to see if we get 180 30 + 150 = 180 180 = 180 Hence, 30 and 150 are linear pair.

  8. Example 3: Find the value of x in the given figure. Solution: Given: From the figure , ∠BCD = 45° and ∠DCA = x ∠BCD and ∠DCA are adjacent angles on a straight line. We know that sum of the adjacent angles on a straight line is 180° ∠BCD + ∠DCA = 180° 45° + x = 180° x = 180° - 45° x = 135° Therefore, the value of x is 135° Since ∠BCA = 180° is a straight angle

  9. Example 4: Find the value of x in the given figure. Solution: Given: From the figure , ∠BCD = 40° , ∠DCE = x and ∠ECA = 30° ∠BCD , ∠DCE and ∠ECA are adjacent angles on a straight line. We know that sum of the adjacent angles on a straight line is 180° ∠BCD + ∠DCA + ∠ECA = 180° 40° + x + 30° = 180° 70° + x = 180° x = 180° - 70° x = 110° Therefore, the value of x is 110° Since ∠BCA = 180° is a straight angle

  10. Try these 1. Find the value of x in the given figure 2. Find the value of x in the given figure

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