Question

Given: BD ⊥ BC ; ∠ABD ≅ ∠DBE Prove: ∠ABD and ∠EBC are complementary

Answer 1

Using the given that BD is perpendicular to BC and that angles ABD and DBE are congruent, it can be proved that ∠ABD and ∠EBC are complementary by showing that their measures add up to 90 degrees.

Step-by-step explanation:

The question requires the use of geometrical theorems and properties to prove that angles ∠ABD and ∠EBC are complementary.

1. First, it is given that BD ⊥ BC and ∠ABD ≅ ∠DBE. If BD is perpendicular to BC, that means ∠BDE is a right angle and measures 90 degrees.
2. Since angles ∠ABD and ∠DBE are congruent, they are of equal measure.
3. Therefore, if we let the measure of ∠ABD be x, then the measure of ∠DBE is also x because of their congruence.
4. Considering that the sum of the angles around a point is 360 degrees, we have ∠ABD (x) + ∠DBE (x) + ∠EBC (y) = 180 degrees.
5. By simplification, 2x + y = 180 degrees. Since ∠BDE is a right angle, x + x = 90 degrees.
6. Subtracting 2x from both sides, we are left with y = 90 degrees - 2x, which means ∠EBC is the complement of 2x, proving that angles ∠ABD and ∠EBC are complementary.