To prove that vertical angles are always congruent using the two statements \( a + b = 180 \) and \( a + c = 180 \), we can follow these steps:
Step 1: Understand the setup
Vertical angles are formed when two lines intersect. Let's denote the angles formed at the intersection as follows:
- Angle \( a \) and angle \( b \) are two angles that are a linear pair (they are adjacent and their non-common sides form a straight line).
- Angle \( c \) is the vertical angle to angle \( a \), and angle \( d \) is the vertical angle to angle \( b \). It follows that \( c \) and \( d \) are also formed at the intersection.
Step 2: Set up the equations
From the problem, we have:
- \( a + b = 180 \) (since angles \( a \) and \( b \) are supplementary)
- \( a + c = 180 \) (since angles \( a \) and \( c \) are also supplementary)
Step 3: Equate the expressions
Using the two equations, we can express them as follows:
\[ b = 180 - a \quad \text{(from the first equation)} \] \[ c = 180 - a \quad \text{(from the second equation)} \]
Step 4: Conclude the congruence
Since both \( b \) and \( c \) equal \( 180 - a \), we can set them equal to each other:
\[ b = c \]
Thus, we conclude that:
\[ \angle b \cong \angle c \]
This shows that vertical angles \( a \) and \( c \) are congruent.
Final statement
Therefore, we have proved that vertical angles are always congruent by using the equations that define their relationships as supplementary angles. Thus, \( \angle b \cong \angle c \) confirms the congruence of vertical angles.
This completes the proof.
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