Question

reasoning in the diagram, ( mangle abd = mangle cbe ). show that ( mangle 1 = mangle 3 ).

Explanation:

Step1: Analyze angle composition

From the diagram, we know that \( m\angle ABD = m\angle 1 + m\angle 2 \) and \( m\angle CBE = m\angle 2 + m\angle 3 \).

Step2: Use given angle equality

We are given that \( m\angle ABD = m\angle CBE \). So we can substitute the angle compositions: \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \).

Step3: Subtract \( m\angle 2 \) from both sides

Subtract \( m\angle 2 \) from each side of the equation \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \). By the Subtraction Property of Equality, we get \( m\angle 1 + m\angle 2 - m\angle 2 = m\angle 2 + m\angle 3 - m\angle 2 \), which simplifies to \( m\angle 1 = m\angle 3 \).

Answer:

By analyzing the angle compositions (\( m\angle ABD = m\angle 1 + m\angle 2 \), \( m\angle CBE = m\angle 2 + m\angle 3 \)), using the given \( m\angle ABD = m\angle CBE \), and applying the Subtraction Property of Equality (subtracting \( m\angle 2 \) from both sides), we conclude \( m\angle 1 = m\angle 3 \).