Question

complete the proof below
given: ( mangle4 = mangle6 )
prove: ( angle arbcongangle crm )

1.) ( mangle4 = mangle6 )
2.) ( mangle arb = mangle4 + mangle5 )( mangle crm = mangle6 + mangle5 )
3.) ( mangle arb - mangle5 = mangle4 )( mangle crm - mangle5 = mangle6 )
4.) ( mangle arb - mangle5 = mangle crm - mangle5 )
5.) ( angle arb=angle crm )
6.) ( angle arbcongangle crm )

Explanation:

Step1: Given

Given \( m\angle4 = m\angle6\)

Step2: Angle - addition postulate

By the angle - addition postulate, \(m\angle ARB=m\angle4 + m\angle5\) and \(m\angle CRM=m\angle6 + m\angle5\)

Step3: Subtraction property of equality

From \(m\angle ARB=m\angle4 + m\angle5\), we get \(m\angle ARB−m\angle5=m\angle4\) (subtraction property of equality). From \(m\angle CRM=m\angle6 + m\angle5\), we get \(m\angle CRM−m\angle5=m\angle6\) (subtraction property of equality)

Step4: Substitution

Since \(m\angle4 = m\angle6\), by substitution \(m\angle ARB−m\angle5=m\angle CRM−m\angle5\)

Step5: Addition property of equality

Add \(m\angle5\) to both sides of \(m\angle ARB−m\angle5=m\angle CRM−m\angle5\). We get \(m\angle ARB=m\angle CRM\) (addition property of equality)

Step6: Definition of congruent angles

If \(m\angle ARB=m\angle CRM\), then \(\angle ARB\cong\angle CRM\) (definition of congruent angles: two angles are congruent if their measures are equal)

Answer:

1. Given; 2. Angle - addition postulate; 3. Subtraction property of equality; 4. Substitution; 5. Addition property of equality; 6. Definition of congruent angles