Main content

## High school geometry

### Course: High school geometry > Unit 3

Lesson 4: Theorems concerning triangle properties# Properties of congruence and equality

Learn when to apply the reflexive property, transitive, and symmetric properties in geometric proofs. Learn the relationship between equal measures and congruent figures.

There are lots of ways to write proofs, and some are more formal than others. In very formal proofs, we justify statements that may feel obvious to you. The reason we justify them is that those claims only work with certain types of relations. What's true with the equality relation isn't necessarily true with the inequality relation, for example.

Let's look at some of these properties. We'll use the symbol \bigstar to represent an unknown relation.

## Reflexive property

When a relation \bigstar has a reflexive property, it means that the relation is always true between a thing and itself. So A, \bigstar, A.

### What are some relations that use it?

Relation | Symbols | Example |
---|---|---|

Equality | equals | minus, 5, start fraction, 3, divided by, 8, end fraction, equals, minus, 5, start fraction, 3, divided by, 8, end fraction |

Congruence | \cong | angle, M, N, P, \cong, angle, M, N, P |

Similarity | \sim | triangle, M, N, P, \sim, triangle, M, N, P |

We use the reflexive property a lot when we're looking at shapes that share sides or angles.

If we were talking about how triangle, M, N, Q and triangle, P, N, Q relate, we might state that start overline, N, Q, end overline, \cong, start overline, N, Q, end overline because of the reflexive property.

### What are some relations that don't?

Strict inequalities don't have a reflexive property. For example, 3, \nless, 3.

Being somebody's mother isn't a reflexive relationship. I am not my own mother.

## Symmetric property

When a relation \bigstar has a symmetric property, it means that the if relation is true between two things, it is true in either order. If A, \bigstar, B, then B, \bigstar, A.

### What are some relations that use it?

Relation | Symbols | Example |
---|---|---|

Equality | equals | If 8, equals, 11, minus, 3, then 11, minus, 3, equals, 8. |

Congruence | \cong | If start overline, V, W, end overline, \cong, start overline, X, Y, end overline, then start overline, X, Y, end overline, \cong, start overline, V, W, end overline. |

Similarity | \sim | If A, B, C, D, \sim, L, M, N, P, then L, M, N, P, \sim, A, B, C, D. |

Parallelism | \parallel | If line m, \parallel line n, then line n, \parallel line m. |

Perpendicularity | \perp | If S, T, with, \overrightarrow, on top, \perp, U, V, with, \overleftrightarrow, on top, then U, V, with, \overleftrightarrow, on top, \perp, S, T, with, \overrightarrow, on top. |

By most people's definitions, friendship is a symmetric relationship. If Alaia is friends with Kolton, then Kolton is friends with Alaia.

### What are some relations that don't?

Strict inequalities don't have a symmetric property. For example, 10, is less than, 100, but 100, \nless, 10.

Being somebody's mother also isn't a symmetric relationship. If Karin is Santino's mother, then Santino cannot be Karin's mother.

## Transitive property

When a relation \bigstar has a transitive property, then two things that relate to a common middle thing also relate to each other. If A, \bigstar, B and B, \bigstar, C, then A, \bigstar, C.

### What are some relations that use it?

Relation | Symbols | Example |
---|---|---|

Equality | equals | If m, angle, F, equals, m, angle, G and m, angle, G, equals, m, angle, H, then m, angle, F, equals, m, angle, H. |

Congruence | \cong | If triangle, R, S, T, \cong, triangle, W, X, Y and triangle, W, X, Y, \cong, triangle, F, G, H, then triangle, R, S, T, \cong, triangle, F, G, H. |

Similarity | \sim | If circle A, \sim circle B and circle B, \sim circle D, then circle A, \sim circle D. |

Parallelism | \parallel | If start overline, J, K, end overline, \parallel, start overline, L, M, end overline and start overline, L, M, end overline, \parallel, start overline, N, O, end overline, then start overline, J, K, end overline, \parallel, start overline, N, O, end overline. |

### What are some relations that don't?

Perpendicularity is not transitive.

In the figure, start overline, A, B, end overline, \perp, start overline, A, C, end overline and start overline, A, C, end overline, \perp, start overline, C, D, end overline, but start overline, A, B, end overline is parallel to, not perpendicular to, start overline, C, D, end overline.

Friendship is also not transitive. If Ezekiel is friends with Romina, and Romina is friends with Nash, we don't know whether or not Ezekiel is friends with Nash.

## Equality versus congruence

Equality and congruence are closely connected, but different. We use equality relations for anything we can express with numbers, including measurements, scale factors, and ratios.

Value | Example |
---|---|

Angle measurements | m, angle, A, plus, m, angle, B, equals, 90, degree |

Segment lengths | M, N, equals, P, Q, equals, 5 |

Area | Area D, E, F, G, equals, 81, start text, c, m, end text, squared |

Ratio | start fraction, 3, divided by, 4, end fraction, equals, start fraction, J, K, divided by, K, L, end fraction |

We use congruence and similarity relations for geometric figures. We can't perform arithmetic operations like addition and multiplication on geometric figures.

Figure | Example |
---|---|

Angle | angle, A, \cong, angle, C |

Line segment | start overline, M, N, end overline, \cong, start overline, P, Q, end overline |

Polygon | triangle, D, E, F, \sim, triangle, G, H, I |

Circle | All circles are similar to all other circles. |

There are three very useful theorems that connect equality and congruence.

- Two angles are congruent if and only if they have equal measures.
- Two segments are congruent if and only if they have equal measures.
- Two triangles are congruent if and only if all corresponding angles and sides are congruent.

So in the following figure, we're given that A, B, equals, C, D, equals, 3, point, 2.

In a very formal proof, we would need a separate line to claim start overline, A, B, end overline, \cong, start overline, C, D, end overline. More casual proofs use equal measures and congruent parts interchangeably. Check with your class to see which you need!

## Want to join the conversation?

- How do we know the difference between equality and congruence(19 votes)
- numbers are equal to each other, and shapes are congruent to each other (same size and shape). Generally, if two angles, as examples, are congruent, then their measures are equal. If two quadrilaterals are congruent, the matching angles and matching sides would all have to be the same measure. Thus, shapes are congruent (because they usually on not directly on top of eache other) just means that matching pairs of sides and angles are all congruent.(15 votes)

- This is ez

How: Try explaining this to a 4 yr old and you'll understand.

Reflexive property: JKL is congruent to JKL

Symmetric property: If JKL is congruent to PQR, then PQR is congruent to JKL.

Transitive property: If JKL is congruent to PQR, and PQR is congruent to XYZ, then JKL is congruent to XYZ.(13 votes) - Why wont the voices stop(8 votes)
- How is your day(6 votes)
- what is difference between equality and congruence in proofs(5 votes)
- Equality is when two things like varibles or measurements are the same numerically:

21 / 3 = 9 - 2 because when you evaluate it they're the same number (7).

Congruence is when to geometric figures like lines, angles, and polygons are the same geometrically:

Triangle ABC is congruent to triangle XYZ because they have all the same side lengths and angle measures, which basically means they're the same shape.(6 votes)

- I hope everyone is understanding his/her work well(6 votes)
- this is just a joke chat(2 votes)
- This is a newer video, so vote up the one question you think is the best and get it to the top. So far, there is not a lot of substance to the chats.(5 votes)

- How do you know if the proof is true for EVERY triangle or just one specific triangle?(3 votes)
- any triangle has 3 sides and 3 angles if those are congruent then the triangle is congruent doesn't matter what kind of triangle they are(3 votes)

- my brain cant process this :((3 votes)
- I wish mine could(2 votes)

- substitution vs. transitive(3 votes)