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### 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 $★$ to represent an unknown relation.

## Reflexive property

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

### What are some relations that use it?

RelationSymbolsExample
Equality$=$$-5\frac{3}{8}=-5\frac{3}{8}$
Congruence$\cong$$\mathrm{\angle }MNP\cong \mathrm{\angle }MNP$
Similarity$\sim$$\mathrm{△}MNP\sim \mathrm{△}MNP$
We use the reflexive property a lot when we're looking at shapes that share sides or angles.
If we were talking about how $\mathrm{△}MNQ$ and $\mathrm{△}PNQ$ relate, we might state that $\stackrel{―}{NQ}\cong \stackrel{―}{NQ}$ 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 $★$ has a symmetric property, it means that the if relation is true between two things, it is true in either order. If $A★B$, then $B★A$.

### What are some relations that use it?

RelationSymbolsExample
Equality$=$If $8=11-3$, then $11-3=8$.
Congruence$\cong$If $\stackrel{―}{VW}\cong \stackrel{―}{XY}$, then $\stackrel{―}{XY}\cong \stackrel{―}{VW}$.
Similarity$\sim$If $ABCD\sim LMNP$, then $LMNP\sim ABCD$.
Parallelism$\parallel$If line $m\parallel$ line $n$, then line $n\parallel$ line $m$.
Perpendicularity$\perp$If $\stackrel{\to }{ST}\perp \stackrel{↔}{UV}$, then $\stackrel{↔}{UV}\perp \stackrel{\to }{ST}$.
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, $10<100$, 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 $★$ has a transitive property, then two things that relate to a common middle thing also relate to each other. If $A★B$ and $B★C$, then $A★C$.

### What are some relations that use it?

RelationSymbolsExample
Equality$=$If $m\mathrm{\angle }F=m\mathrm{\angle }G$ and $m\mathrm{\angle }G=m\mathrm{\angle }H$, then $m\mathrm{\angle }F=m\mathrm{\angle }H$.
Congruence$\cong$If $\mathrm{△}RST\cong \mathrm{△}WXY$ and $\mathrm{△}WXY\cong \mathrm{△}FGH$, then $\mathrm{△}RST\cong \mathrm{△}FGH$.
Similarity$\sim$If circle $A\sim$ circle $B$ and circle $B\sim$ circle $D$, then circle $A\sim$ circle $D$.
Parallelism$\parallel$If $\stackrel{―}{JK}\parallel \stackrel{―}{LM}$ and $\stackrel{―}{LM}\parallel \stackrel{―}{NO}$, then $\stackrel{―}{JK}\parallel \stackrel{―}{NO}$.

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

Perpendicularity is not transitive.
In the figure, $\stackrel{―}{AB}\perp \stackrel{―}{AC}$ and $\stackrel{―}{AC}\perp \stackrel{―}{CD}$, but $\stackrel{―}{AB}$ is parallel to, not perpendicular to, $\stackrel{―}{CD}$.
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 measurements$m\mathrm{\angle }A+m\mathrm{\angle }B=90\mathrm{°}$
Segment lengths$MN=PQ=5$
AreaArea $DEFG=81\phantom{\rule{0.167em}{0ex}}{\text{cm}}^{2}$
Ratio$\frac{3}{4}=\frac{JK}{KL}$
We use congruence and similarity relations for geometric figures. We can't perform arithmetic operations like addition and multiplication on geometric figures.
FigureExample
Angle$\mathrm{\angle }A\cong \mathrm{\angle }C$
Line segment$\stackrel{―}{MN}\cong \stackrel{―}{PQ}$
Polygon$\mathrm{△}DEF\sim \mathrm{△}GHI$
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 $AB=CD=3.2$.
In a very formal proof, we would need a separate line to claim $\stackrel{―}{AB}\cong \stackrel{―}{CD}$. 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
• 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.
• what is difference between equality and congruence in proofs
• 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.
• How do you know if the proof is true for EVERY triangle or just one specific triangle?
• 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
• What is the difference between equality and congruency?
• "Equal" means "these 'two' things are actually one thing with two names". We say 1+4=3+2 because '1+4' is a description of a certain number, and '3+2' is a description of the same exact number. Equality is a broad concept used across, and outside, all of math.

Congruency is a concept specific to geometry. We say two figures are congruent if you can set one perfectly on top of the other without distorting either of them. More rigorously, if you can translate, rotate, and/or reflect one figure so that it lands perfectly on the other figure, then the two figures are congruent.

If we say two triangles are congruent, it means they have the same side lengths, angles, and area, but they may be in different locations, tilted with respect to each other, or oriented differently. If we say triangle ABC is equal to triangle DEF, it means the triangles already fully coincide; maybe there is a single point that is named both A and D, a single point named both B and E, and a single point named both C and F.
• How do we know the difference between equality and congruence
(1 vote)
• There is a slight difference between congruence and equality. Congruence relates segments, angles, and figures, whereas equality relates numbers, which can include lengths of segments and measures of angles. For example, if angles 1 and 2 have the same measure, we would say that angle 1 is congruent to angle 2, whereas we would say that the measure of angle 1 equals the measure of angle 2.
• How do you find the sum to 180
• Draw line a through points A and B. Draw line b through point C and parallel to line a. Since lines a and b are parallel, <)BAC = <)B'CA and <)ABC = <)BCA'. It is obvious that <)B'CA + <)ACB + <)BCA' = 180 degrees.
(1 vote)
• if given a quadrilateral that is not given to be a parallelogram, how do we prove the figure are congruent without numbers
• Use a proven postulate.
• How do you tell the symbols apart?
(1 vote)
• are these articles extensions? Are they optional if you want to learn more or something
(1 vote)
• in the last part: "In a very formal proof, we would need a separate line to claim AB=CD. More casual proofs use equal measures and congruent parts interchangeably. Check with your class to see which you need!", what does this separate line look like?
(1 vote)
• It's talking about two-column proofs. For the problem in question we have:

AB = CD = 3.2-----Given

This means that AB ≅ CD, because two segments are congruent if they have equal measures, but we still have to point that out. What the article is getting at is that in very formal proofs, you can never use any information that is not either given or proven, so we would have to change it to this:

AB = CD = 3.2-----Given
AB ≅ CD-----Two segments with equal measures are congruent.
(1 vote)