Question

complete the proof below
given: ( mangle 4 = mangle 6 )
prove: ( angle arbcongangle crm )
statement
1.) ( mangle 4 = mangle 6 )
2.) ( mangle arb = mangle 4 + mangle 5 )
( mangle crm = mangle 6 + mangle 5 )
3.) ( mangle arb - mangle 5 = mangle 4 )
( mangle crm - mangle 5 = mangle 6 )
4.) ( mangle arb - mangle 5 = mangle crm - mangle 5 )
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: Solve for \(m\angle4\) and \(m\angle6\)

Subtract \(m\angle5\) from both equations: \(m\angle ARB - m\angle5=m\angle4\) and \(m\angle CRM - m\angle5=m\angle6\)

Step4: Substitution

Since \(m\angle4 = m\angle6\), substitute \(m\angle4\) with \(m\angle6\) (or vice - versa) in the equations from Step 3. So, \(m\angle ARB - m\angle5=m\angle CRM - m\angle5\)

Step5: Subtraction property of equality

Add \(m\angle5\) to both sides of the equation \(m\angle ARB - m\angle5=m\angle CRM - m\angle5\). By the subtraction property of equality (if \(a - c=b - c\), then \(a = b\)), we get \(m\angle ARB=m\angle CRM\)

Step6: Definition of congruent angles

If the measures of two angles are equal (\(m\angle ARB=m\angle CRM\)), then by the definition of congruent angles (\(\angle A\cong\angle B\) if and only if \(m\angle A=m\angle B\)), \(\angle ARB\cong\angle CRM\)

Answer:

The proof is completed as shown above, with each step justified by the given information, angle - addition postulate, substitution, subtraction property of equality, and the definition of congruent angles.