Question

given: ( mangle orp = 80^{circ} )( mangle orn=(3x + 10)^{circ} )prove: ( x = 30 )which statement could be used in step 2 when proving ( x = 30 )?( angle orp ) and ( angle orn ) are a linear pair( angle orp ) and ( angle orn ) are vertical angles( 80 = 3x+10 )( x = 30 )

Explanation:

Step1: Recall linear pair property

A linear pair of angles is supplementary, i.e., their sum is \(180^{\circ}\).

Step2: Analyze each option

• Option 1: If \(\angle ORP\) and \(\angle ORN\) are a linear pair, then \(m\angle ORP + m\angle ORN=180^{\circ}\). This is a valid geometric property and can be used as a statement in the proof.
• Option 2: Vertical angles are equal. But \(\angle ORP\) and \(\angle ORN\) are not vertical angles (vertical angles are opposite angles formed by two intersecting lines).
• Option 3: \(80 = 3x + 10\) is an equation that would come after using the linear - pair property (if \(\angle ORP\) and \(\angle ORN\) are a linear pair, then \(80+(3x + 10)=180\), and then simplifying gives \(80 = 3x + 10\) after subtracting 100 from both sides of \(80+(3x + 10)=180\)). It is not a basic geometric property statement for step 2.
• Option 4: \(x = 30\) is what we need to prove, not a statement for step 2.

Answer:

\(\angle ORP\) and \(\angle ORN\) are a linear pair.