Congruent 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.”
The flipped triangle is still congruent. Tick marks and arcs show which parts match, and the letter order in △ ABC ≅ △ DEF says the same thing.
Formal definition and notation
Two figures are congruent if a sequence of rigid motions (rotations, reflections and translations) carries one onto the other. Segments are congruent if and only if their lengths are equal, and angles are congruent if and only if their measures are equal. Two polygons are congruent if and only if all pairs of corresponding sides and all pairs of corresponding angles are congruent. In a congruence statement such as △ ABC ≅ △ DEF, the order of the letters tells which vertices correspond: A with D, B with E, C with F.
on a diagram, sides with the same number of ticks are congruent, and angles with the same number of arcs are congruent
Key facts
Congruent describes size and shape, not position. A figure that is rotated or reflected is still congruent to the original.
Use = for numbers and ≅ for figures: write AB = 5 and AB = CD, but AB ≅ CD.
In △ ABC ≅ △ DEF, matching letters sit in matching positions, so ∠ B ≅ ∠ E and AC ≅ DF.
Corresponding parts of congruent triangles are congruent (often shortened to CPCTC), so once triangles are proven congruent, every matching side and angle is equal.
Congruent figures have equal perimeter and equal area, but equal area alone does not make figures congruent: a 2 by 8 and a 4 by 4 rectangle both have area 16.
Congruence rules for triangles
Congruence rules for triangles
SSS, SAS, ASA, AAS ⇒ congruent triangles
SSS
all three pairs of sides are congruent
SAS
two pairs of sides and the angle between them are congruent
ASA
two pairs of angles and the side between them are congruent
AAS
two pairs of angles and a side that is not between them are congruent
Any one of these four is enough to prove two triangles congruent. AAA and SSA are not: AAA gives similar triangles of any size, and SSA can fit two different triangles.
Try it: flip it and compare
Interactive
Choose Reflect, then switch the line of reflection. The image turns over but every side and angle keeps its size, so it stays congruent. Then try Dilate to see what breaks congruence.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Congruent examples, step by step
Example 1: Name the congruent parts
Warm-up
Given △ JKL ≅ △ PQR, which angle is congruent to ∠ K, and which side is congruent to JL?
Line up the letters in order: J with P, K with Q, L with R.
∠ K is in the 2nd position, so it matches the 2nd letter of PQR: ∠ Q.
JL uses the 1st and 3rd letters, so it matches PR.
Answer:∠ K ≅ ∠ Q and JL ≅ PR.
Example 2: Use congruence to find a length
Core
△ ABC ≅ △ XYZ. AB = 7 cm, BC = 10 cm and XZ = 12 cm. Find YZ and the perimeter of △ ABC.
YZ matches BC (2nd and 3rd letters), so YZ = 10 cm.
XZ matches AC, so AC = 12 cm.
Perimeter of △ ABC = 7 + 10 + 12 = 29 cm.
Answer:YZ = 10 cm and the perimeter of △ ABC is 29 cm.
Example 3: Solve for x with congruent angles
Test-style
In the figure, ∠ M ≅ ∠ N. If m∠ M = (3x + 14)° and m∠ N = (5x − 6)°, what is m∠ M?
“Congruent figures” is the topic of whole shapes that match; “congruent” is the adjective and the symbol you use for any pair of segments, angles or figures, as in ∠ 1 ≅ ∠ 2.
Similar figures have the same shape but may differ in size; congruent figures match in shape and size. Every congruent pair is similar with scale factor 1.
Congruentvs.Equal
Equal compares numbers, such as lengths or angle measures (AB = 6); congruent compares the figures themselves (AB ≅ CD).
Quick check: 3 questions
Question 1 of 3
If △ RST ≅ △ UVW, which side is congruent to ST?
Show answer and explanation
Answer: B. S and T are the 2nd and 3rd letters, which match V and W, so ST ≅ VW. UV matches RS, UW matches RT, and TS is the same segment, not a match in the other triangle.
Question 2 of 3
Two angles are congruent. One measures (2x + 30)° and the other (4x − 10)°. What is x?
Show answer and explanation
Answer: A. Set the measures equal: 2x + 30 = 4x − 10, so 40 = 2x and x = 20. 70 is the angle measure, not x; 10 and 20/3 come from adding or subtracting the wrong terms.
Question 3 of 3
Which information is NOT enough to prove two triangles congruent?
Show answer and explanation
Answer: C. AAA only fixes the shape, so the triangles could be different sizes (similar, not congruent). SSS, SAS and ASA each include enough sides to fix the size too.
Congruent: frequently asked questions
What is the symbol for congruent?
The symbol is ≅, an equal sign with a tilde on top. The tilde stands for same shape and the equal sign for same size. Read △ ABC ≅ △ DEF as “triangle ABC is congruent to triangle DEF.” On a diagram, congruence is shown with matching tick marks on sides and matching arcs on angles.
Are congruent shapes the same as equal shapes?
In everyday talk, yes, but in geometry we say figures are congruent and their measurements are equal. Two segments of length 5 are congruent, and their lengths are equal: AB ≅ CD and AB = CD = 5. Congruent figures can sit in different places and face different directions.
Can a reflected shape be congruent?
Yes. Reflections, rotations and translations are rigid motions: they move a figure without stretching or shrinking it, so the image is always congruent to the original. Only a dilation with a scale factor other than 1 changes the size, which breaks congruence but keeps the figures similar.
RotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Geometry · Grade 8, GeometryReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometryTranslationA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Geometry · Grade 8, GeometryLengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3PolygonA 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 5SideA 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 2Corresponding 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, GeometryEqual SignThe equal sign (=) is the symbol that shows two expressions have the same value. In 3 + 5 = 2 + 6, both sides equal 8, so the statement is true. The equal sign does not mean “the answer comes next”; it works like a balanced scale, and a number sentence with = is called an equation.Symbols and Notation · Kindergarten, Grade 1, Grade 2Similar FiguresSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Geometry · Grade 7, Grade 8, Algebra 1NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2TriangleA 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 5Congruent FiguresCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Geometry · Grade 7, Grade 8PerimeterPerimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.Geometry · Grade 3, Grade 4, Grade 5AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4, Grade 5SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6TransformationA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Geometry · Grade 7, Grade 8, Algebra 1DilationA dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center point. Each point moves along the ray from the center to k times its distance. With center at the origin, (x, y) → (kx, ky). The image keeps its shape, so it is similar, not congruent.Geometry · Grade 8, GeometryScale FactorA scale factor is the number used to multiply side lengths when making a similar figure larger or smaller. A scale factor of 2 doubles every length, and a scale factor of 12 makes lengths half as large. Find it by dividing a new length by the matching original length. Areas change by the square of the scale factor.Ratios and Proportions · Grade 7, Grade 8, Algebra 1