Choose a mode and enter the angles you know.
Enter numbers only. For angle sizes, decimals and fractions such as 250/7 are OK.
Result and figure
Choose a mode on the left, enter the angle and press "Calculate". The result and a figure of the parallel lines will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- In the classic figure of two parallel lines cut by a transversal, enter one angle \(\angle a\) and get all eight angles \(\angle a\) to \(\angle h\). Angles that do not divide evenly, such as \(\dfrac{250}{7}^\circ\), are shown both as an exact fraction and as a decimal
- A color-coded figure and table show which pairs are corresponding angles, alternate interior angles, vertical angles and same-side interior angles (angles of the same color are equal)
- In the bent line mode, it finds the angle at the bend of a bent line between two parallel lines, \(\angle x = \angle a + \angle b\), with steps for the standard method: draw an auxiliary line through the bend parallel to both lines and split the angle into alternate interior angles
- Formula cards with figures explain why corresponding angles and alternate interior angles are equal, why vertical angles are equal, and why same-side interior angles add up to \(180^\circ\)
How the angles are labeled differs between textbooks (many US textbooks number them ∠1 to ∠8). On this page, the angles at the upper intersection are ∠a, ∠b, ∠c and ∠d, going counterclockwise from the upper right, and the angles at the lower intersection are ∠e, ∠f, ∠g and ∠h in the same order (you can see the positions in the figure with the result).
What is this calculation used for?
Measuring the size of the Earth (Eratosthenes and surveying)
Sunlight comes from so far away that its rays can be treated as parallel on the ground. About 2,200 years ago, the Greek scholar Eratosthenes knew that on a certain day the sun was directly overhead in the town of Syene. On that day, in Alexandria, far to the north, he measured the angle between the sun's rays and a vertical stick from its shadow (about 7.2°). Because alternate interior angles of parallel lines are equal, this 7.2° is the same as the angle between the two towns seen from the center of the Earth. 7.2° is exactly 1/50 of 360°, so he multiplied the distance between the towns by 50 and estimated the distance around the Earth quite accurately.
Measuring the size of the Earth long before satellites, this is still the most famous use of parallel lines and alternate interior angles in textbooks.
Drawing parallel lines by sliding a triangle (drafting and art)
Hold a drafting triangle against a straightedge and slide it along, drawing lines in the same direction as you go: all of those lines are parallel. As the triangle slides, its angle (the angle between the straightedge and the line you draw, a corresponding angle) does not change, so the rule "if corresponding angles are equal, the lines are parallel" becomes a drawing method.
The parallel rule and the drafting machine used in architectural and mechanical drafting work on the same principle, keeping a straightedge parallel as it moves up and down.
Cutting stair balusters at the same angle (carpentry and construction)
A stair handrail is one sloped line, and the balusters that hold it up are many vertical, parallel lines. Because corresponding angles of parallel lines are equal, the angle between the handrail and each baluster is the same, so the tops of all the balusters can be cut at the same angle.
This is why the same part can be cut in quantity without fitting each one by hand.
The path of light in a periscope (optical devices)
A periscope, familiar from submarines and science projects, has two mirrors at the top and bottom of a tube, each tilted at 45° and facing each other in parallel. Light turned by the upper mirror is turned again by the lower mirror and reaches your eye. Because the two mirrors are parallel, the light coming in and the light going out are parallel through alternate interior angles, and the view is not tilted.
The path of light, bending twice between parallel lines, looks much like the figure in the bent line mode on this page (the actual direction of light is set by the law of reflection).
Formulas and figures
Vertical angles are equal (even without parallel lines)
Figure
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\(\angle a\)
\(=\)
\(\angle c\)
①
\(\angle a\): one angle
\(=\)
②
\(\angle c\): the opposite angle (vertical angle)
The formula in words
①
When two lines cross,
\(\angle a\): one angle
②
is always equal to
\(\angle c\): the opposite angle (vertical angle)
Quick example
When one of the angles formed by two crossing lines is \(60^\circ\), its vertical angle is
vertical angle \(\angle c\)
\(=\)
\(\angle a\) (\(60^\circ\))
\(\angle c = \angle a = 60^\circ\)
Key idea
Why are they equal? Both \(\angle a\) and \(\angle c\) make a straight \(180^\circ\) when added to the angle between them, \(\angle b\) (\(\angle a = 180^\circ - \angle b\) and \(\angle c = 180^\circ - \angle b\)). They are both the same number subtracted from \(180^\circ\), so they are always the same size.
The vertical angle rule works whenever two lines cross. Parallel lines have nothing to do with it. This is the difference from corresponding and alternate interior angles, and it is often tested.
Corresponding angles of parallel lines are equal
Figure
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\(\angle a\)
\(=\)
\(\angle e\)
①
\(\angle a\): one angle
\(=\)
②
\(\angle e\): its corresponding angle
The formula in words
①
When lines \(\ell\) and \(m\) are parallel,
\(\angle a\): one angle
②
is equal to
\(\angle e\): the angle in the same position (corresponding angle)
Quick example
When a transversal crosses two parallel lines and \(\angle a = 60^\circ\), its corresponding angle \(\angle e\) is
corresponding angle \(\angle e\)
\(=\)
\(\angle a\) (\(60^\circ\))
\(\angle e = \angle a = 60^\circ\)
Key idea
Corresponding angles are a pair of angles in the same position at the two intersections. If \(\angle a\) is at the upper right of the upper intersection, its corresponding angle \(\angle e\) is at the upper right of the lower intersection.
Corresponding angles are equal only when the two lines are parallel. If the lines are not parallel, they are not equal.
The converse is also true: if corresponding angles are equal, the two lines must be parallel. This two-way rule, "parallel, so the angles are equal" and "the angles are equal, so parallel", is the core of the parallel lines unit.
Alternate interior angles of parallel lines are equal
Figure
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\(\angle c\)
\(=\)
\(\angle e\)
①
\(\angle c\): one interior angle
\(=\)
②
\(\angle e\): its alternate interior angle
The formula in words
①
When lines \(\ell\) and \(m\) are parallel,
\(\angle c\): one interior angle
②
is equal to
\(\angle e\): the interior angle on the other side (alternate interior angle)
Quick example
When a transversal crosses two parallel lines and the interior angle \(\angle c = 60^\circ\), its alternate interior angle \(\angle e\) is
alternate interior angle \(\angle e\)
\(=\)
\(\angle c\) (\(60^\circ\))
\(\angle e = \angle c = 60^\circ\)
Key idea
Alternate interior angles are a pair of angles inside the parallel lines (between the two lines) and on opposite sides of the transversal. The two angles and the transversal look like the letter Z, so the "Z shape" is a well-known way to spot them.
Why are they equal? \(\angle c\) is the vertical angle of \(\angle a\), so \(\angle c = \angle a\). And \(\angle a\) and \(\angle e\) are corresponding angles, so \(\angle a = \angle e\). Put together, \(\angle c = \angle a = \angle e\). It follows from just the vertical angle rule and the corresponding angle rule.
Like corresponding angles, if alternate interior angles are equal, the two lines are parallel (a converse theorem).
Same-side interior angles of parallel lines add up to 180°
Figure
Standard notation
(the usual math form)
\(\angle c\)
\(+\)
\(\angle f\)
\(=\)
\(180^\circ\)
In words
(symbols replaced with words)
①
\(\angle c\): one interior angle
\(+\)
②
\(\angle f\): the interior angle on the same side
\(=\)
③
\(180^\circ\): straight angle
The formula in words
①
When lines \(\ell\) and \(m\) are parallel, add
\(\angle c\): one interior angle
②
and
\(\angle f\): the interior angle on the same side of the transversal (same-side interior angle)
③
and you always get
\(180^\circ\): straight angle
Quick example
When a transversal crosses two parallel lines and one same-side interior angle is \(\angle c = 60^\circ\), the other one, \(\angle f\), is
same-side interior angle \(\angle f\)
\(=\)
\(180^\circ\)
\(-\)
\(\angle c\) (\(60^\circ\))
\(180^\circ - 60^\circ = 120^\circ\)
Key idea
Same-side interior angles (also called consecutive interior angles) are a pair of angles inside the parallel lines and on the same side of the transversal. Alternate interior angles are "on opposite sides and equal", while same-side interior angles are "on the same side and add up to \(180^\circ\)" (they are supplementary).
Why \(180^\circ\)? \(\angle f\) and \(\angle e\) are adjacent angles on a straight line, so \(\angle f = 180^\circ - \angle e\). By alternate interior angles, \(\angle e = \angle c\). So \(\angle c + \angle f = \angle e + (180^\circ - \angle e) = 180^\circ\).
This is exactly why, in a trapezoid (a quadrilateral with one pair of parallel sides), two neighboring angles between the parallel sides add up to \(180^\circ\).
Angle of a bent line between parallel lines
Figure
Standard notation
(the usual math form)
\(\angle x\)
\(=\)
\(\angle a\)
\(+\)
\(\angle b\)
In words
(symbols replaced with words)
③
\(\angle x\): angle at the bend
\(=\)
①
\(\angle a\): angle at the upper line
\(+\)
②
\(\angle b\): angle at the lower line
The formula in words
①
When a line bends between two parallel lines, add
\(\angle a\): angle at the upper line
②
and
\(\angle b\): angle at the lower line
③
and you get
\(\angle x\): angle at the bend
Quick example
When the upper angle is \(\angle a = 35^\circ\) and the lower angle is \(\angle b = 25^\circ\), the angle at the bend \(\angle x\) is
angle at the bend \(\angle x\)
\(=\)
\(\angle a\) (\(35^\circ\))
\(+\)
\(\angle b\) (\(25^\circ\))
\(\angle x = 35^\circ + 25^\circ = 60^\circ\)
Key idea
The key to solving it is an auxiliary line through the bend, parallel to the two lines. Draw it and split \(\angle x\) into an upper part and a lower part. The upper part is the alternate interior angle of \(\angle a\), and the lower part is the alternate interior angle of \(\angle b\), so \(\angle x = \angle a + \angle b\).
"Find the angle between parallel lines" is a classic problem in Geometry class. Even with two or more bends, the same idea works: draw a parallel auxiliary line through each bend.
This formula works when \(\angle a + \angle b\) is less than \(180^\circ\) (at exactly \(180^\circ\), the line no longer bends and becomes straight).
When a transversal crosses two parallel lines, four rules decide every angle: vertical angles are equal (even without parallel lines), corresponding angles are equal, alternate interior angles are equal, and same-side interior angles add up to \(180^\circ\). If you know one angle, the eight angles split into four that equal it and four that equal \(180^\circ\) minus it. For a line that bends between parallel lines, draw an auxiliary line through the bend parallel to both lines, and \(\angle x = \angle a + \angle b\).
Symbols and terms
Symbols
| \(\angle\) | angle | The symbol for an angle. \(\angle a\) is "the angle named \(a\)". When three letters are used, as in \(\angle APB\), the middle letter is the vertex of the angle. |
| \(\ell,\ m\) | ell, m | Names for lines. \(\ell\) is a script form of the letter l, from "line", written this way so it is not confused with the number 1 or a capital I. The next lines are named \(m\), \(n\) and so on in alphabetical order. |
| \(\parallel\) | is parallel to | The symbol for parallel. \(\ell \parallel m\) says "line \(\ell\) is parallel to line \(m\)". It is said to come from drawing two parallel lines as two vertical strokes. |
| \(x\) | x | A letter for an unknown number (the angle you want to find). Calling an unknown \(x\) is a familiar habit in textbooks. |
| \(^\circ\) | degree | The symbol for the degree, the unit of angle. One full turn divided into 360 equal parts is \(1^\circ\). The number 360 is said to have been chosen because it has many factors and is easy to divide evenly. |
Terms
| parallel lines | Two lines in the same plane that never meet, however far they are extended. They are written \(\ell \parallel m\). The distance (width) between two parallel lines is the same wherever you measure it. |
| transversal | A line that crosses two other lines. Where a transversal crosses two parallel lines, it makes four angles at each of the two intersections, eight angles in all. |
| vertical angles | Of the four angles formed when two lines cross, a pair of angles opposite each other. Vertical angles are always equal. This holds whenever two lines cross, whether or not any lines are parallel. |
| corresponding angles | When a transversal crosses two lines, a pair of angles in the same position at the two intersections (for example, both at the upper right). If the two lines are parallel, corresponding angles are equal. Conversely, if corresponding angles are equal, the two lines are parallel. |
| alternate interior angles | A pair of angles inside the two lines (between them) and on opposite sides of the transversal. If the two lines are parallel, alternate interior angles are equal. The two angles and the transversal look like a Z, so the "Z shape" is a well-known way to spot them. |
| same-side interior angles | A pair of angles inside the two lines and on the same side of the transversal (also called consecutive interior angles). If the two lines are parallel, same-side interior angles add up to \(180^\circ\). The two neighboring angles between the parallel sides of a trapezoid have exactly this relationship. |
| auxiliary line | A line you add to a figure yourself to solve a problem. For the angle of a bent line between parallel lines, the key step is an auxiliary line through the bend, parallel to the two lines. |
| converse theorems | The theorems that go the other way: "if corresponding angles are equal, the lines are parallel" and "if alternate interior angles are equal, the lines are parallel". They are used to prove that two lines are parallel from the sizes of angles. |
| straight angle (linear pair) | A straight angle is half a turn, \(180^\circ\). Around a point on a straight line, the angles on one side always add up to \(180^\circ\); two adjacent angles that do this form a linear pair. This is the basis of "adjacent angles add up to \(180^\circ\)" and "vertical angles are equal". |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to the topics in this list is the quickest way forward.
| Angles and turns (Grade 4) |
|
| Perpendicular and parallel lines (Grade 4) |
|
| Straight angles and supplementary angles (Grades 4–7) |
|
| Expressions with letters (Grades 6–7) |
|
| Subtraction and fractions (Grades 5–6) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find all 8 angles from one angle
| Angle ∠a (degrees) | 60 |
| ∠c, ∠e, ∠g (equal to ∠a) (degrees) | =B1 |
| ∠b, ∠d, ∠f, ∠h (degrees) | =180-B1 |
Table to find the angle of a bent line
| Angle at the upper line ∠a (degrees) | 35 |
| Angle at the lower line ∠b (degrees) | 25 |
| Angle at the bend ∠x (degrees) | =B1+B2 |
After pasting, the upper rows are your inputs and the lower rows are calculated automatically.
"-" is subtraction and "+" is addition.
The first table enters 60 for ∠a. The angles equal to ∠a through vertical, corresponding and alternate interior angles (∠c, ∠e, ∠g) are 60, and the other four angles (∠b, ∠d, ∠f, ∠h) are 180 − 60 = 120.
The second table is the bent line example. Enter 35 for ∠a and 25 for ∠b, and the angle at the bend ∠x is 35 + 25 = 60.
"-" is subtraction and "+" is addition.
The first table enters 60 for ∠a. The angles equal to ∠a through vertical, corresponding and alternate interior angles (∠c, ∠e, ∠g) are 60, and the other four angles (∠b, ∠d, ∠f, ∠h) are 180 − 60 = 120.
The second table is the bent line example. Enter 35 for ∠a and 25 for ∠b, and the angle at the bend ∠x is 35 + 25 = 60.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find all 8 angles from one angle
| Angle ∠a (degrees) | 60 |
| ∠c, ∠e, ∠g (equal to ∠a) (degrees) | =B1 |
| ∠b, ∠d, ∠f, ∠h (degrees) | =180-B1 |
Table to find the angle of a bent line
| Angle at the upper line ∠a (degrees) | 35 |
| Angle at the lower line ∠b (degrees) | 25 |
| Angle at the bend ∠x (degrees) | =B1+B2 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the angles with your own numbers.
How to calculate it in Python
from fractions import Fraction
# Parallel lines and a transversal: all 8 angles from one angle
angle_a = Fraction(60) # a fraction such as 250/7° can be written Fraction(250, 7)
angle_b = 180 - angle_a # adjacent angle (subtract from the straight angle 180°)
print(f"∠a = ∠c = ∠e = ∠g = {angle_a}° (equal as vertical, corresponding and alternate interior angles)")
print(f"∠b = ∠d = ∠f = ∠h = {angle_b}°")
print(f"Check same-side interior angles: {angle_a}° + {angle_b}° = {angle_a + angle_b}°")
# Bent line between parallel lines: the angle at the bend
bend_a = Fraction(35)
bend_b = Fraction(25)
angle_x = bend_a + bend_b
print(f"Angle at the bend ∠x = {bend_a}° + {bend_b}° = {angle_x}°")
With the fractions module from the standard library, angles that do not divide evenly, such as 250/7°, stay exact fractions with no rounding error. When you run it, the first part shows that the angles split into four angles of 60° and four of 120°, and that same-side interior angles add up to 180°. The second part shows the angle at the bend, 60°. Change the numbers and run it.
How to write it in LaTeX and other math languages (copy and paste)
Vertical angles are equal (even without parallel lines)
∠a = ∠c
\angle a = \angle c
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mo>∠</mo><mi>a</mi>
<mo>=</mo>
<mo>∠</mo><mi>c</mi>
</mrow>
</math>
/_ a = /_ c
angleA == angleC
angleA = angleC;
angleA == angleC
∠a = ∠c
Corresponding angles of parallel lines are equal
∠a = ∠e
\angle a = \angle e
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mo>∠</mo><mi>a</mi>
<mo>=</mo>
<mo>∠</mo><mi>e</mi>
</mrow>
</math>
/_ a = /_ e
angleA == angleE
angleA = angleE;
angleA == angleE
∠a = ∠e
Alternate interior angles of parallel lines are equal
∠c = ∠e
\angle c = \angle e
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mo>∠</mo><mi>c</mi>
<mo>=</mo>
<mo>∠</mo><mi>e</mi>
</mrow>
</math>
/_ c = /_ e
angleC == angleE
angleC = angleE;
angleC == angleE
∠c = ∠e
Same-side interior angles of parallel lines add up to 180°
∠c + ∠f = 180°
\angle c + \angle f = 180^{\circ}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mo>∠</mo><mi>c</mi>
<mo>+</mo>
<mo>∠</mo><mi>f</mi>
<mo>=</mo>
<msup><mn>180</mn><mo>°</mo></msup>
</mrow>
</math>
/_ c + /_ f = 180^circ
angleC + angleF == 180 Degree
angleC + angleF = 180;
angleC + angleF == 180
∠c + ∠f = 180°
Angle of a bent line between parallel lines
∠x = ∠a + ∠b
\angle x = \angle a + \angle b
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mo>∠</mo><mi>x</mi>
<mo>=</mo>
<mo>∠</mo><mi>a</mi>
<mo>+</mo>
<mo>∠</mo><mi>b</mi>
</mrow>
</math>
/_ x = /_ a + /_ b
angleX == angleA + angleB
angleX = angleA + angleB;
angleX == angleA + angleB
∠x = ∠a + ∠b
How to have ChatGPT do the calculation
You are a math (plane geometry, parallel lines and angles) calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). For the eight angles formed when a transversal crosses two parallel lines, show each of the following: 1. The sizes of all eight angles when one angle is 250/7° (also group the equal angles; give each as a fraction in lowest terms and as a decimal) 2. A check that same-side interior angles add up to 180° 3. For a line that bends between the parallel lines, the angle at the bend when the angle at the upper line is 35° and the angle at the lower line is 25° In Python, use the fractions module from the standard library to calculate exactly, and show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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