Question

given: ( mangle eyg = mangle hxk )( mangle 2 = mangle 4 )prove: ( mangle 1 = mangle 3 )

Explanation:

Step1: Angle addition postulate

By the angle addition postulate, \(m\angle HXK=m\angle3 + m\angle4\) (same as \(m\angle EYG=m\angle1 + m\angle2\) which is given by the angle addition postulate).

Step2: Given information

Since \(m\angle EYG = m\angle HXK\) is given.

Step3: Substitution property

Substitute \(m\angle EYG=m\angle1 + m\angle2\) and \(m\angle HXK=m\angle3 + m\angle4\) into \(m\angle EYG = m\angle HXK\), we get \(m\angle1 + m\angle2=m\angle3 + m\angle4\) (substitution property).

Step4: Given information

Since \(m\angle2 = m\angle4\) is given.

Step5: Subtraction property of equality

Subtract \(m\angle2\) (or \(m\angle4\)) from both sides of \(m\angle1 + m\angle2=m\angle3 + m\angle4\). Using the subtraction property of equality: \(m\angle1 + m\angle2-m\angle2=m\angle3 + m\angle4 - m\angle4\), so \(m\angle1=m\angle3\).

Answer:

The reasons for statement 2 is "Angle Addition Postulate", for statement 3 is "Given", for statement 4 is "Substitution Property", for statement 5 is "Given".