Question

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

Explanation:

Step1: Angle Addition Postulate

By the Angle Addition Postulate, if two non - overlapping angles form a larger angle, the measure of the larger angle is the sum of the measures of the two smaller angles. So, \(m\angle QPS=m\angle1 + m\angle2\) (Angle Addition Postulate) and \(m\angle VPT=m\angle3 + m\angle4\) (Angle Addition Postulate).

Step2: Given

We are given that \(m\angle QPS = m\angle VPT\).

Step3: Substitution

Since \(m\angle QPS=m\angle1 + m\angle2\), \(m\angle VPT=m\angle3 + m\angle4\), and \(m\angle QPS = m\angle VPT\), by substitution, \(m\angle1 + m\angle2=m\angle3 + m\angle4\).

Step4: Given

We are given that \(m\angle2 = m\angle4\).

Step5: Substitution

Substitute \(m\angle2\) with \(m\angle4\) in the equation \(m\angle1 + m\angle2=m\angle3 + m\angle4\). We get \(m\angle1 + m\angle4=m\angle3 + m\angle4\).

Step6: Subtraction Property of Equality

Subtract \(m\angle4\) from both sides of the equation \(m\angle1 + m\angle4=m\angle3 + m\angle4\). Using the Subtraction Property of Equality (\(a + b=a + c\Rightarrow b = c\) when \(a\) is common), we have \(m\angle1=m\angle3\).

Answer:

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