Question

use the given information to prove that ( mangle2 = mangle4 ).
given: ( mangle abc = mangle edc )
( mangle1 = mangle3 )
prove: ( mangle2 = mangle4 )

Explanation:

Step1: Use the angle - addition postulate

By the angle - addition postulate, \(m\angle ABC=m\angle1 + m\angle2\) and \(m\angle EDC=m\angle3 + m\angle4\).

Step2: Substitute the given equal angles

Since \(m\angle ABC = m\angle EDC\) (given) and \(m\angle1 = m\angle3\) (given), we substitute into the equations from Step 1: \(m\angle1 + m\angle2=m\angle3 + m\angle4\).

Step3: Use the subtraction property of equality

Subtract \(m\angle1\) (which is equal to \(m\angle3\)) from both sides of the equation \(m\angle1 + m\angle2=m\angle3 + m\angle4\). Using the subtraction property of equality (\(a + b=a + c\) implies \(b = c\) when \(a\) is non - zero), we get \(m\angle2=m\angle4\).

Answer:

\(m\angle2 = m\angle4\)