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 )?
statements | reasons
1. ( mangle orp = 80^circ ), ( mangle orn = (3x + 10)^circ ) | 1. given
2. | 2.
3. | 3.
4. | 4.
5. | 5.
( angle orp ) and ( angle orn ) are a linear pair
( angle orp ) and ( angle orn ) are vertical angles
( 80 = 3x + 10 )
( x = 30 )

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 )?
statements | reasons
1. ( mangle orp = 80^circ ), ( mangle orn = (3x + 10)^circ ) | 1. given
2.  | 2.
3.  | 3.
4.  | 4.
5.  | 5.
( angle orp ) and ( angle orn ) are a linear pair
( angle orp ) and ( angle orn ) are vertical angles
( 80 = 3x + 10 )
( x = 30 )

Accepted Answer

# Brief Explanations:
To prove $ x = 30 $, we first analyze the angles. From the diagram, $ \angle ORP $ and $ \angle ORN $ form a linear pair (they are adjacent and form a straight line, so their sum is $ 180^\circ $). But in step 2, we need to establish the relationship between the angles. Wait, actually, looking at the diagram, $ \angle ORP $ and $ \angle ORN $ are supplementary? No, wait, the diagram shows that $ \angle ORP = 80^\circ $ and $ \angle ORN=(3x + 10)^\circ $, and they are on a straight line (since $ P - R - N $ is a straight line? Wait, no, the diagram has $ P $, $ R $, $ N $ on a horizontal line? Wait, no, the angles at $ R $: $ \angle ORP = 80^\circ $ and $ \angle ORN=(3x + 10)^\circ $, and also there's a vertical line $ O - R - M $. Wait, actually, $ \angle ORP $ and $ \angle ORN $ are supplementary? No, wait, the correct relationship: if $ \angle ORP $ and $ \angle ORN $ are a linear pair, their sum is $ 180^\circ $, but in the answer choices, the first option is " $ \angle ORP $ and $ \angle ORN $ are a linear pair". Wait, but let's check the other options: vertical angles are equal, but these are not vertical angles. The third option is $ 80 = 3x + 10 $, which would be if they are equal, but that's not the case. The fourth option is $ x = 30 $, which is the conclusion, not step 2. Wait, maybe I misread the diagram. Wait, the diagram shows that $ \angle ORP = 80^\circ $ and $ \angle ORN=(3x + 10)^\circ $, and they are adjacent angles forming a linear pair? Wait, no, maybe $ \angle ORP $ and $ \angle ORN $ are supplementary? Wait, no, the correct step 2 should be stating that $ \angle ORP $ and $ \angle ORN $ are a linear pair (so their sum is $ 180^\circ $), but wait, the answer choices: the first option is " $ \angle ORP $ and $ \angle ORN $ are a linear pair". Wait, but let's check the angles. Wait, maybe the diagram is such that $ \angle ORP $ and $ \angle ORN $ are a linear pair, so their sum is $ 180^\circ $, but in the answer choices, the first option is that. Wait, but let's re - evaluate. The problem is to find which statement is used in step 2. The given is step 1: $ m\angle ORP = 80^\circ $, $ m\angle ORN=(3x + 10)^\circ $. Step 2 should be the relationship between the angles. If $ \angle ORP $ and $ \angle ORN $ are a linear pair, then $ m\angle ORP + m\angle ORN=180^\circ $, which would lead to $ 80+(3x + 10)=180 $, and solving gives $ 3x+90 = 180 $, $ 3x = 90 $, $ x = 30 $. But the answer choices: the first option is " $ \angle ORP $ and $ \angle ORN $ are a linear pair". Wait, but let's check the other options. Vertical angles are equal, but these are not vertical angles. The third option is $ 80 = 3x + 10 $, which would be if they are equal, but that's not the case. The fourth option is the conclusion. So the correct statement for step 2 is that $ \angle ORP $ and $ \angle ORN $ are a linear pair. Wait, but wait, maybe the diagram is different. Wait, looking at the diagram, $ P $, $ R $, $ N $ are on a horizontal line? No, the diagram has $ P $ to the left of $ R $, $ N $ to the right of $ R $, and $ O $ below $ R $, $ M $ above $ R $, and $ L $ at an angle. Wait, maybe $ \angle ORP $ and $ \angle ORN $ are supplementary? No, maybe they are equal? No, that doesn't make sense. Wait, maybe I made a mistake. Wait, the correct answer is the first option: " $ \angle ORP $ and $ \angle ORN $ are a linear pair". Because that establishes the relationship between the two angles, allowing us to set up the equation in step 3.
# Answer:
$\boldsymbol{\angle ORP}$ and $\boldsymbol{\angle ORN}$ are a linear pair

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