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(1) ∠ABC = 65°, ∠DBE = 65°, ∠CBE = 115°, ∠ABD = 115°

(2) ∠ABC = 62°, ∠DBE = 62°, ∠CBE = 118°, ∠ABD = 118°

Solution:

(1) In the given image ABC and DBE are vertical angles.

Vertical angle theorem:

If two angles are vertical then they are congruent.

⇒ ∠ABC = ∠DBE

⇒ 3x° + 38° = 5x° + 20°

Arrange like terms one side.

⇒ 38° – 20° = 5x° – 3x°

⇒ 18° = 2x°

x° = 9°

∠ABC = 3(9°) + 38° = 65°

∠DBE = 5(9°) + 20° = 65°

Adjacent angles in a straight line = 180°

⇒ ∠ABC + ∠CBE = 180°

⇒ 65° + ∠CBE = 180°

∠CBE = 115°

∠ABD and ∠CBE are vertical angles.

∠ABD = 115°

(2) In the given image ABC and DBE are vertical angles.

⇒ ∠ABC = ∠DBE

⇒ 4x° + 2° = 5x° – 13°

Arrange like terms one side.

⇒ 13° + 2° = 5x° – 4x°

⇒ 15° = x°

∠ABC = (4(15°) + 2°) = 62°

∠DBE = 5(15°) – 13° = 62°

Adjacent angles in a straight line = 180°

⇒ ∠ABC + ∠CBE = 180°

⇒ 62° + ∠CBE = 180°

∠CBE = 118°

∠ABD and ∠CBE are vertical angles.

∠ABD = 118°


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