Key takeaway: Corresponding angles are two angles in the same position at two different crossings of a transversal, and they are equal when the lines being crossed are parallel.
Grade 88.G.A.5Geometry and Measurement
Definition of Corresponding Angles
Corresponding angles sit in matching corners when a transversal crosses two lines. One angle is at the first crossing and the other is at the second, and both are on the same side of the transversal and on the same side (above or below) of their lines.
Think of sliding the top crossing down the transversal until it lands on the bottom crossing: they are the ones that land on top of each other. When the two lines are parallel, corresponding angles are equal. One equal pair of them forces the two lines to be parallel.
Formula and notation
Number the angles so that \(\angle 2\) is at the top crossing and \(\angle 6\) is in the same corner at the bottom crossing. Then:
\[a \parallel b \ \Rightarrow\ m\angle 2 = m\angle 6\]
Four pairs exist: upper-left with upper-left, upper-right with upper-right, and so on for the lower corners.
Key facts
- Every transversal and pair of lines gives four pairs of corresponding angles.
- The pair often forms the letter F, turned or flipped.
- Equal matching-corner angles show that the two lines are parallel.
- If the lines are not parallel, corresponding angles have different measures.
Corresponding Angles example
A transversal crosses two parallel fences. The angle at the upper right of the first crossing is \((3x + 22)^{\circ}\), and the corresponding angle at the second crossing is \((5x – 36)^{\circ}\). Find \(x\).
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Parallel lines make corresponding angles equal: \(3x + 22 = 5x – 36\).
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Move terms: \(58 = 2x\), so \(x = 29\).
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Check: \(3(29) + 22 = 109\) and \(5(29) – 36 = 109\).
\(x = 29\), and both angles measure \(109^{\circ}\).
Common mistake
Watch out
Leo pairs the upper-right angle at the top with the lower-left angle at the bottom and calls them corresponding, so he sets them equal.
Why it is wrong
Corresponding means the same corner at each crossing. Upper-right goes with upper-right. The upper-right angle and the lower-left angle on opposite ends are alternate exterior angles, which are also equal for parallel lines, but a different name and position. Always name the corner for both angles before you write the equation.
Related terms and lessons
- Grade 8 · Angles, Parallel Lines and TrianglesLessonWorksheetQuiz
- Geometry · Parallel Lines, Transversals and ProofsLessonWorksheetQuiz
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