Question

given: ( mangle qrs = mangle uts )( mangle 1 = mangle 3 )prove: ( mangle 2 = mangle 4 )
1 ( mangle qrs = mangle 1 + mangle 2 ) reason?
2 ( mangle uts = mangle 3 + mangle 4 ) reason?
3 ( mangle qrs = mangle uts ) reason?
4 ( mangle 1 + mangle 2 = mangle 3 + mangle 4 ) reason?
5 ( mangle 1 = mangle 3 ) reason?
6 ( mangle 3 + mangle 2 = mangle 3 + mangle 4 ) reason?
7 ( mangle 2 = mangle 4 ) reason?

Explanation:

Step1: Angle Addition Postulate

The Angle Addition Postulate states that if \( \angle AOB\) and \( \angle BOC\) are adjacent angles, then \( m\angle AOC=m\angle AOB + m\angle BOC\). For \( \angle QRS\), since \( \angle 1\) and \( \angle 2\) are adjacent and form \( \angle QRS\), \( m\angle QRS=m\angle 1 + m\angle 2\). Similarly, for \( \angle UTS\), since \( \angle 3\) and \( \angle 4\) are adjacent and form \( \angle UTS\), \( m\angle UTS=m\angle 3 + m\angle 4\).

Step2: Given

We are directly given in the problem statement that \( m\angle QRS = m\angle UTS\).

Step3: Substitution

Substitute \( m\angle QRS=m\angle 1 + m\angle 2\) and \( m\angle UTS=m\angle 3 + m\angle 4\) into \( m\angle QRS = m\angle UTS\). So, \( m\angle 1 + m\angle 2=m\angle 3 + m\angle 4\).

Step4: Given

We are directly given in the problem statement that \( m\angle 1 = m\angle 3\).

Step5: Substitution

Substitute \( m\angle 1\) with \( m\angle 3\) in the equation \( m\angle 1 + m\angle 2=m\angle 3 + m\angle 4\). We get \( m\angle 3 + m\angle 2=m\angle 3 + m\angle 4\).

Step6: Subtraction Property of Equality

If \( a + b=a + c\), then \( b = c\) (subtract \( a\) from both sides). Here, subtract \( m\angle 3\) from both sides of \( m\angle 3 + m\angle 2=m\angle 3 + m\angle 4\).

Answer:

1. Angle Addition Postulate
2. Angle Addition Postulate
3. Given
4. Substitution (from 1,2,3)
5. Given
6. Substitution (from 4,5)
7. Subtraction Property of Equality