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## Chapter 5: Chapter 5

Problem 60

Expert verified
Given that $$∠ABC \cong ∠DBE$$, we can apply the Angle Sum Property of Triangles to triangles ABE and DBE. After equating the sum of the interior angles of both triangles and simplifying the equation by using the given congruent angles, we can conclude that $$∠ABD \cong ∠CBE$$.
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## Step 1: State the given information.

We are given that in triangle ABE, $$∠ABC \cong ∠DBE$$.

## Step 2: Identify angle properties of triangles.

In any triangle, the sum of the interior angles is equal to 180°. This is known as the Angle Sum Property of Triangles.

## Step 3: Apply the angle properties to triangle ABE.

Since the sum of the interior angles in triangle ABE is 180°, we have: $∠ABC + ∠ABE + ∠BAE = 180°$

## Step 4: Identify angle properties of triangle DBE.

Again, using the Angle Sum Property of Triangles, we have the sum of the interior angles in triangle DBE equal to 180°: $∠DBE + ∠EBD + ∠BDA = 180°$

## Step 5: Apply given congruent angles and subtract from both equations.

We are given that $$∠ABC \cong ∠DBE$$. Using this information, we can equate the two angle expressions from Steps 3 and 4: $∠ABC + ∠ABE + ∠BAE = ∠DBE + ∠EBD + ∠BDA$ Now, let's subtract $$∠DBE$$ from both sides to isolate the angles we want to prove congruent: $∠ABC + ∠ABE + ∠BAE - ∠DBE = ∠EBD + ∠BDA$

## Step 6: Simplify and prove angles congruent.

Since $$∠ABC \cong ∠DBE$$, their measures are equal. Additionally, since $$∠BAE = ∠BDA$$, we can replace these terms in the simplified equation: $∠ABE + ∠BAE = ∠EBD + ∠BDA$ Hence, we can conclude that $$∠ABD \cong ∠CBE$$.

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