Alternate interior angles are two angles that lie between two lines and on opposite sides of a transversal that crosses them. When the two lines are parallel, alternate interior angles are equal. They often trace a Z shape: in the diagram, angles 3 and 6 form one pair and angles 4 and 5 the other.
Interior means between the two lines; alternate means on opposite sides of the transversal. When the lines are parallel, each alternate interior pair is equal.
Formal definition and notation
Let transversal t cross lines m and n at distinct points. Two angles are alternate interior angles if both lie in the region between m and n, on opposite sides of t, with vertices at different crossings. The Alternate Interior Angles Theorem: if m ∥ n, each pair is congruent. Conversely, if one pair is congruent, then m ∥ n.
Symbol
Read it as
Meaning
∠ 3 ≅ ∠ 6
angle 3 is congruent to angle 6
an alternate interior pair has equal measures when the lines are parallel
Key facts
A transversal across two lines makes exactly 2 pairs of alternate interior angles.
Trace the Z (or a backward Z, or a stretched N): the two angles tucked into its corners are alternate interior angles.
The theorem works backwards: equal alternate interior angles prove the lines are parallel.
In a parallelogram, a diagonal is a transversal of opposite sides, so it makes equal alternate interior angles.
The classic proof that a triangle‘s angles total 180° draws a line parallel to one side and uses two pairs of alternate interior angles.
Alternate Interior Angles Theorem
Alternate Interior Angles Theorem
m ∥ n ⇒ ∠ 3 = ∠ 6 and ∠ 4 = ∠ 5
⇒
implies; read it as if … then
∠ 3, ∠ 6
between the lines, on opposite sides of the transversal
Why: ∠ 3 is vertical to ∠ 2, and ∠ 2 corresponds to ∠ 6, so ∠ 3 = ∠ 6.
Alternate Interior Angles examples, step by step
Example 1: Use the theorem
Warm-up
Lines m ∥ n are cut by a transversal. One alternate interior angle is 48°. Find its partner and the other interior angle at the same crossing.
Alternate interior angles are equal with parallel lines: the partner is 48°.
At one crossing, the two interior angles form a straight line, so the other is 180° − 48° = 132°.
Answer: The partner is 48°, and the other interior angle at that crossing is 132°.
Example 2: Solve for x
Core
m ∥ n. Alternate interior angles measure (9x − 22)° and (5x + 30)°. Find each angle.
Set them equal: 9x − 22 = 5x + 30.
4x = 52, so x = 13.
Angle: 9(13) − 22 = 95°. Check: 5(13) + 30 = 95.
Answer:x = 13 and each angle is 95°.
Example 3: Spot the Z
Test-style
In the figure, AB ∥ DE, and segment BD crosses both. ∠ ABD = 37°. What is ∠ BDE if A and E are on opposite sides of BD?
BD is a transversal of the parallel segments AB and DE.
A and E are on opposite sides of BD, and both angles open between the parallels, so ∠ ABD and ∠ BDE are alternate interior angles.
With parallel lines they are equal: ∠ BDE = 37°.
Answer:∠ BDE = 37°.
Common mistakes
✗ MistakePairing ∠ 3 with ∠ 5 and setting them equal.
✓ Instead∠ 3 and ∠ 5 are same-side interior angles; they add to 180°. ∠ 3‘s alternate partner is ∠ 6.
Both are interior, but they are on the same side of the transversal, so they are supplementary, not equal.
✗ MistakeAssuming alternate interior angles are equal in any figure with a Z shape.
✓ InsteadCheck that the two lines are parallel first. If ∠ 3 = 70° and ∠ 6 = 75°, the lines are not parallel.
The Z shape only identifies the pair. Equality needs parallel lines.
✗ MistakeChoosing two angles outside the parallel lines as alternate interior.
✓ InsteadAngles outside the lines, such as ∠ 1 and ∠ 8, are alternate exterior angles.
Interior means in the strip between the two lines.
Alternate exterior angles are outside the two lines, on opposite sides of the transversal (∠ 1 and ∠ 8). They are also equal when the lines are parallel.
Same-side interior angles are between the lines on the same side of the transversal (∠ 3 and ∠ 5). With parallel lines they add to 180° instead of being equal.
An interior angle of a polygon is an angle inside the shape at a vertex, such as each 108° angle of a regular pentagon. It has nothing to do with a transversal.
Quick check: 3 questions
Question 1 of 3
Lines are parallel. An alternate interior angle is 113°. What is its partner?
Show answer and explanation
Answer: B. Alternate interior angles are equal when the lines are parallel, so the partner is 113°. 67° is the supplement (that would be the same-side interior angle), 23° subtracts 90, and 247° uses 360.
Question 2 of 3
Which statement proves m ∥ n?
Show answer and explanation
Answer: D. Equal alternate interior angles prove the lines parallel. 64° and 116° are unequal, same-side interior angles of 80° add to 160, not 180, and vertical angles are equal at every crossing, so they prove nothing.
Question 3 of 3
m ∥ n. Alternate interior angles are (4x + 6)° and (6x − 30)°. What is each angle?
Show answer and explanation
Answer: B.4x + 6 = 6x − 30 gives x = 18, so each angle is 4(18) + 6 = 78°. 18 is x, not the angle; 102 is the supplement; 84 comes from x = 19.5, an arithmetic slip.
They are congruent when the two lines cut by the transversal are parallel, which is the usual setting in textbook and test problems. If the lines are not parallel, the angles are still called alternate interior angles, but their measures differ. Always look for parallel marks or a stated ∥ before setting them equal.
What is the Z rule for alternate angles?
Draw or spot a letter Z made of the two parallel lines and the transversal. The two acute angles tucked inside the corners of the Z are alternate interior angles, and so are the two obtuse angles of a reversed Z. The Z may be stretched, flipped or turned on its side; the rule still works.
How many pairs of alternate interior angles are there?
Two. A transversal crossing two lines makes four interior angles, two at each crossing. Each interior angle pairs with the one at the other crossing on the opposite side of the transversal: in standard numbering, ∠ 3 with ∠ 6 and ∠ 4 with ∠ 5. With parallel lines, one pair is acute and the other obtuse, unless all are right angles.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.G.A.5, CCSS HSG-CO.C.9
Cite this entry
Effortless Math Team. “Alternate Interior Angles.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/alternate-interior-angles/
Flashcard
TermAlternate Interior Anglesinside the lines, opposite sides: the Z
DefinitionAngles between two lines on opposite sides of a transversal. With parallel lines they are equal: ∠ 3 = ∠ 6 = 55°.
TransversalA transversal is a line that crosses two or more other lines at different points. Crossing two lines makes eight angles, four at each intersection. When the two lines are parallel, those angles come in equal groups: corresponding, alternate interior and alternate exterior angles are equal, and same-side interior angles add to 180 degrees.Geometry · Grade 8, GeometryParallel LinesParallel lines are lines in the same plane that never intersect. They stay the same distance apart and point in the same direction, like the rails of a straight railroad track. On a coordinate plane, parallel lines have the same slope but different y-intercepts. The symbol ∥ means is parallel to.Geometry · Grade 4, Grade 5, Grade 6SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometryParallelogramA parallelogram is a four-sided shape whose opposite sides are parallel. Its opposite sides are also equal in length, its opposite angles are equal, and its diagonals cut each other in half. Rectangles, rhombuses and squares are special parallelograms. Its area is base times height: A = bh.Geometry · Grade 3, Grade 5, Grade 6ProofA proof is a step-by-step logical argument that shows a math statement is true in every case it covers. It starts from given facts and uses definitions, postulates and statements already proved, with a reason for each step. A statement that has been proved is called a theorem.Geometry · Geometry, Algebra 1TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5Interior AngleAn interior angle is an angle inside a polygon, formed at a vertex by two sides that meet there. A polygon with n sides has n interior angles, and they add up to (n − 2) × 180 degrees. A triangle’s interior angles total 180 degrees, and a rectangle’s four right angles total 360 degrees.Geometry · Grade 7, Grade 8, GeometryPolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5Acute AngleAn acute angle is an angle that measures more than 0 degrees and less than 90 degrees. It is narrower than the square corner of a page. Angles of 15, 45 and 89 degrees are all acute; 90 degrees is a right angle, so it is not acute. Every triangle has at least two acute angles.Geometry · Grade 4, Grade 5, Grade 7Obtuse AngleAn obtuse angle is an angle that measures more than 90 degrees and less than 180 degrees. It opens wider than a square corner but does not flatten into a straight line. Angles of 100, 135 and 179 degrees are obtuse. A triangle can have at most one obtuse angle.Geometry · Grade 4, Grade 5, Grade 7Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4TheoremA theorem is a math statement that has been proved true using definitions, postulates and statements already proved. Once proved, it is true in every case it covers, so you can use it to solve problems. The Pythagorean theorem and the Vertical Angles Theorem are famous examples.Geometry · Grade 8, GeometryCorresponding AnglesCorresponding angles are two angles in the same relative position at each place where a transversal crosses two lines, such as the top-left corner at both crossings. When the two lines are parallel, corresponding angles are equal. A transversal across two lines makes four pairs of corresponding angles.Geometry · Grade 8, GeometryVertical AnglesVertical angles are the pair of opposite angles formed when two lines intersect. They share a vertex but no sides, and they are always equal in measure. Two crossing lines make two pairs of vertical angles: if one angle is 120°, the angle across from it is also 120°.Geometry · Grade 7, Grade 8, GeometrySupplementary AnglesSupplementary angles are two angles whose measures add up to 180 degrees. Each one is the supplement of the other, so the supplement of an angle x is 180° − x. When they share a side, they form a straight line, called a linear pair. For example, 118° and 62° are supplementary.Geometry · Grade 7, Grade 8, Geometry