(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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