Question

use the given information to prove that ( mangle1 = mangle2 ).
given: ( mangle abe = mangle deg )
( mangle3 = mangle4 )
prove: ( mangle1 = mangle2 )
2 ( mangle3 = mangle4 )
3 ( mangle abe = mangle1 + mangle3 )
4 ( mangle1 = mangle2 )

Explanation:

Step1: Use the Angle Addition Postulate

$$m\angle ABD=m\angle 1 + m\angle 3$$
$$m\angle KEC=m\angle 2 + m\angle 4$$

Step2: Substitute given equal angles

Since \(m\angle ABD = m\angle KEC\) (given) and \(m\angle 3=m\angle 4\) (given), we substitute into the equations from Step 1:
\(m\angle 1 + m\angle 3=m\angle 2 + m\angle 4\)

Step3: Subtract equal angles

Subtract \(m\angle 3\) (which is equal to \(m\angle 4\)) from both sides of the equation:
\(m\angle 1=m\angle 2\)

Answer:

\(m\angle 1=m\angle 2\)