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## 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?

RelationSymbolsExample
Equalityequalsminus, 5, start fraction, 3, divided by, 8, end fraction, equals, minus, 5, start fraction, 3, divided by, 8, end fraction
Congruence\congangle, M, N, P, \cong, angle, M, N, P
Similarity\simtriangle, 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.
Two triangles M N Q and P N Q share the same side N Q. Side N M is congruent to side N P. Side M Q is congruent to side P Q. Side N Q is congruent to itself.
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?

RelationSymbolsExample
EqualityequalsIf 8, equals, 11, minus, 3, then 11, minus, 3, equals, 8.
Congruence\congIf 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\simIf A, B, C, D, \sim, L, M, N, P, then L, M, N, P, \sim, A, B, C, D.
Parallelism\parallelIf line m, \parallel line n, then line n, \parallel line m.
Perpendicularity\perpIf 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?

RelationSymbolsExample
EqualityequalsIf m, angle, F, equals, m, angle, G and m, angle, G, equals, m, angle, H, then m, angle, F, equals, m, angle, H.
Congruence\congIf 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\simIf circle A, \sim circle B and circle B, \sim circle D, then circle A, \sim circle D.
Parallelism\parallelIf 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.
ValueExample
Angle measurementsm, angle, A, plus, m, angle, B, equals, 90, degree
Segment lengthsM, N, equals, P, Q, equals, 5
AreaArea D, E, F, G, equals, 81, start text, c, m, end text, squared
Ratiostart 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.
FigureExample
Angleangle, A, \cong, angle, C
Line segmentstart overline, M, N, end overline, \cong, start overline, P, Q, end overline
Polygontriangle, D, E, F, \sim, triangle, G, H, I
CircleAll circles are similar to all other circles.
There are three very useful theorems that connect equality and congruence.
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!