Provide reasons for the proof. Given: Line m is parallel to line n. Prove: Angle A is supplementary to Angle B.
Understand the Problem
The question requires a proof in geometry, specifically that a certain angle is supplementary to another angle given that line m is parallel to line n. The task involves understanding the properties of parallel lines and transversal angles.
Answer
$ A + B = 180^\circ $
Answer for screen readers
The angles are supplementary, specifically ( A + B = 180^\circ ).
Steps to Solve
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Identify the Given Angles
Let angle ( A ) and angle ( B ) be formed when a transversal crosses lines ( m ) and ( n ) which are parallel. Identify which angle needs to be proven as supplementary. -
Apply the Parallel Lines Theorem
According to the properties of parallel lines, when a transversal intersects two parallel lines, the alternate interior angles are equal, and the same-side interior angles are supplementary. -
State the Supplementary Condition
For angles to be supplementary, the sum of their measures must equal ( 180^\circ ). Therefore, if angle ( A ) is one angle and angle ( B ) is the angle on the same side of the transversal, we can express this as:
$$ A + B = 180^\circ $$ -
Conclude the Proof
Since ( A ) and ( B ) are on the same side of the transversal and based on the property of supplementary angles formed by a transversal with parallel lines, we have proven that angle ( A ) is supplementary to angle ( B ).
The angles are supplementary, specifically ( A + B = 180^\circ ).
More Information
This property of angles is essential in many geometric proofs and reflects the fundamental theorem of alternate interior angles being equal and same-side interior angles being supplementary when dealing with parallel lines and a transversal.
Tips
- Confusing alternate interior angles with same-side interior angles. Remember, alternate angles are equal, whereas same-side interior angles are supplementary.
- Forgetting to add up the angles to check for supplementarity. Always confirm that the sum equals ( 180^\circ ).
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