Question

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

Explanation:

Step1: Analyze angle composition

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

Step2: Substitute given equality

We know \( m\angle ABD = m\angle CBE \), so 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: \( (m\angle 1 + m\angle 2) - m\angle 2 = (m\angle 2 + m\angle 3) - m\angle 2 \).
Simplifying both sides gives \( m\angle 1 = m\angle 3 \).

Answer:

By analyzing the angle compositions (\( m\angle ABD = m\angle 1 + m\angle 2 \), \( m\angle CBE = m\angle 2 + m\angle 3 \)), substituting the given \( m\angle ABD = m\angle CBE \), and subtracting \( m\angle 2 \) from both sides, we conclude \( m\angle 1 = m\angle 3 \).