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Congruent Triangles

Congruent Triangles

How is being "equal" different from being "congruent" in Geometry? When talking about shapes, we can say that one side is equal to another or the respective angles are equal: this term works well when talking about lengths or another numeric value. If you say, 'These two triangles are equal,' you're not providing the full story. The next questions would be, 'What is equal? The area? The lengths of the sides?' When two or more triangles have all equal areas, angles, and side lengths, we can more precisely describe them as congruent.

Congruent triangles: Meaning and examples

Congruent triangles have the same shape and size, with equal sides and angles, but they can be positioned differently from each other in space. When talking about congruent triangles, there must be two or more triangles in order to compare them with each other. You can't evaluate congruence on one triangle because it will always be congruent to itself! Let's look at an example that compares two triangles.

Imagine you have a right triangle, and your friend is sitting on the opposite side of a table from you with a copy of your triangle. You both put your triangles on the table with the right angle on the left side like this:

Congruent triangles, right triangles congruent, StudySmarterRight triangles turned 180º relative to one another, StudySmarter Original

These two triangles are congruent: they are the same size and shape. With a turn and a drag, the triangles can precisely overlap each other, like so:

Congruent Triangles, congruent triangle explanation, StudySmarterCongruent triangles being dragged onto each other, StudySmarter Originals

When two or more triangles can overlap each other exactly, we know that they are congruent.

Given the example above, can you define congruent triangles? Try comparing your definition with the following one:

Congruent triangles are triangles of the same shape and size. However, they can be positioned differently in space.

Let's see a different example.

Three triangles are positioned differently from each other. One of them is also oriented differently; that is, it is rotated relative to the others. Just by looking at them, do you think these triangles are congruent? Take a look at the given triangles in the picture below.

Congruent Triangles Differently positioned triangles StudySmarterDifferently positioned triangles, StudySmarter Original

The first two triangles on top look congruent, right? They are the same shape and size as each other. The third triangle on the bottom may look a bit different from the first two because of the way it's oriented. If you were to rotate the third triangle 70° clockwise, you could more easily see that it is in fact congruent to the other two: it's the same shape and size. See the picture below.

Congruent Triangles, congruent triangle, StudySmarterCongruent triangles with the same orientation, StudySmarter Original

From this example, we see that triangles can still be congruent even if they are rotated or oriented differently in space. The same goes for congruent triangles that are flipped (reflected) or slid over (translated).

Notations for congruent triangles

We know that two triangles can be congruent, but now the question is: "How can we tell which side or angle from each triangle corresponds with which?" So, let's see how equal sides and angles are marked and identified.

Typically, we mark equal sides of congruent triangles with dash-like lines, while the angles have curved markings over them. We can see these notations in the figure of congruent triangles below. Notice that different corresponding sides and angles have their own matching notations to show which one matches with which. To avoid mixing up the separate sides and angles, different pairs have differing numbers of marks (i.e., one line, two lines, three lines).

Congruent Triangles, congruent triangle, StudySmarterCongruent triangles, StudySmarter Originals

In the above figure, sides AB and DE are equal, so it is marked by a single dash. Similarly, sides BC and EF are equal, and sides AC and DF are equal. Also, which is shown by a single curved mark. Double and triple curved marks are used as a notation to show that and , respectively. Note that the sign is used to show that the triangles are congruent. For the above figure, we can say that ; that is, triangle ABC is congruent to triangle DEF.

Congruent triangles versus non-congruent triangles

It's important to note that for congruent triangles to stay congruent, we can only perform transformations of translation (location change) or rotation on any one of them. If we need to transform the shape or size (or some angles or lengths) of a triangle to make them exactly overlap one another, then the triangles are non-congruent. Let's define non-congruent triangles.

Non-congruent triangles are triangles differing in shape and/or size relative to each other.

Let's look at an example.

Two triangles are given in the picture below. Do they look congruent?

Congruent Triangles, non-congruent triangle, StudySmarterTwo non-congruent triangles, StudySmarter Originals

Solution: This case is pretty evident: the given triangles are different in shape and size, no matter how you move or rotate them. This means the given triangles are non-congruent.

Check whether the given triangles are congruent or not.

Congruent Triangles, similar triangle, StudySmarterTwo similar triangles, StudySmarter Originals

Solution: Based on the triangles' angle notations, we can see that By looking at the figure, it may seem as though the triangles are congruent. However, the triangle ABC is bigger than the triangle DEF. Although the triangles have the same shape, they are not congruent. When we drag triangle DEF onto triangle ABC, they don't fit each other exactly. This is because their sides' lengths are not equal.

Congruent Triangles, similar triangle, StudySmarterSimilar triangles placed on each other, StudySmarter Originals

Triangle DEF fits inside triangle ABC. Hence, the triangles have the same shape but different sizes, making them non-congruent triangles. They are, however, similar!

Similar triangles are triangles that have the same shape (due to their equal corresponding angles) but are different in size.

Rules for determining congruent triangles

Now that we know the concept of congruent triangles, how can we determine if triangles are congruent? In the first two examples, we dragged and turned the triangles to see if they were congruent. You may be wondering if we can tell if triangles are congruent just by looking at them closely. However, relying on vision alone can be faulty and inaccurate. Rotating and dragging the triangles is also not the best method! There are more accurate methods for finding out if triangles are congruent.

As you might have guessed, we can determine congruency by measuring the triangles. Those with equal corresponding sides and angles are congruent. Let's see an example of this method.

Two triangles are positioned right next to each other. The second one is rotated. We need to find out whether these two are congruent.

Congruent Triangles, different oriented congruent triangles, StudySmarterDifferently oriented triangles, StudySmarter Original

First, let's rotate the second triangle, so it's positioned in the same orientation as the first triangle. This step is not mandatory, but it helps to see the shapes more clearly and compare the right sides and angles.

Congruent Triangles, equally oriented congruent triangles, StudySmarterEqually oriented triangles, StudySmarter Originals

Next, let's measure the angles and sides of both triangles. To understand our results more clearly, let's name the triangles ABC and DEF.

Congruent Triangles named congruent triangles StudySmarterNamed triangles, StudySmarter Original

The measurements for the given triangles are:

By comparing the triangles' measurements, we can see that they are all equal. This means the triangles are congruent. We denote this with the following symbol:

It would take quite some time to measure every side and every angle whenever we wanted to determine triangle congruency. For this reason, we rely on triangle congruency theorems which help us reduce the number of measurements needed to evaluate congruency.

Congruent triangles types and theorems

Suppose you only know some information about two triangles' measurements, such as the measurements of two of their angles and the side length between those angles. If these specific measurements are equal between the two triangles, this limited amount of information is enough to prove that they are congruent. Why don't we need all of the triangles' angle measurements and side lengths to confirm it? This is because we can refer to the known triangle congruency theorem, Angle-Side-Angle (ASA), which states that our equal measurements (two angles and their shared side) are sufficient.

Depending on the type of information that we have about the triangles' measurements, we can select from the five triangle congruency theorems to help us evaluate congruency. The known theorems for triangle congruence are shown in the table below. We can also think of these theorems as shortcuts of measurement, each with their own specific rules and conditions.

Theorem NameStatementFigure
Side-Side-Side(SSS)The triangles are congruent if... all three sides of one triangle are equal to all three sides of another triangle.

Congruent Triangles SSS Theorem StudySmarterSSS congruent triangles, StudySmarter Originals

Side-Angle-Side(SAS)The triangles are congruent if... two sides and the included angle of these sides of one triangle are equal to the two sides and included angle of another triangle.

Congruent Triangles SAS Theorem StudySmarterSAS congruent triangles, StudySmarter Originals

Hypothenuse-Leg(HL)The triangles are congruent if... the hypothenuse side and any one side of one right triangle are equal to the hypotenuse and another side of another right triangle.

Congruent Triangles, HL congruent triangle, StudySmarterHL congruent triangles, StudySmarter Originals

Angle-Side-Angle(ASA)The triangles are congruent if... two angles and the included side of one triangle are equal to the two angles and included side of another triangle.

Congruent Triangles ASA Theorem StudySmarterASA congruent triangles, StudySmarter Originals

Angle-Angle-Side(AAS)The triangles are congruent if... two angles and a non-included side of one triangle are equal to the two angles and a non-included side of another triangle.

Congruent Triangles AAS Theorem StudySmarterAAS congruent triangles, StudySmarter Originals

Congruent triangles - Key takeaways

  • Triangles are congruent if their respective sides and angles are equal (they have the same shape and size).
  • Congruent triangles can be placed in different locations and rotated relative to each other.
  • Non-congruent triangles differ in shape and/or size.
  • There are rules or theorems that help determine whether triangles are congruent or not.

Frequently Asked Questions about Congruent Triangles

When talking about congruence, it usually involves two or more triangles. If two or more triangles are congruent, it means they have the same shape and size - equal sides and angles.

To prove that two or more triangles are congruent, you need to know that the given triangles have equal respective sides and angles. You can also use one of the five theorems for proving triangle congruence (SSS, SAS, ASA, AAS and HL).

To find congruent triangles, you need to look at the shape and size of the given triangles. The triangles that have equal respective sides and angles are congruent. You can also use one of the five theorems to find congruent triangles (SSS, SAS, ASA, AAS and HL).

The rules for congruent triangles are - they need to have equal respective sides and angles. These two rules are also encompassed in five theorems for proving triangle congruence - SSS, SAS, ASA, AAS and HL.

An example of congruent triangles is two equilateral triangles with a side of length 6cm. Equilateral triangles have all sides of the same length, so this means two such triangles with a side of the same length are congruent - all of the sides and angles are equal between the two triangles.

Final Congruent Triangles Quiz

Question

What are congruence transformations?

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Answer

Congruence transformations are transformations which when performed on an object produce congruent objects.

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Question

What are the 3 types of congruence transformations?

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Answer

  • Rotation

  • Reflection

  • Translation

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Question

How can you visually identify a congruence transformation?


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Answer

A congruence transformation does not change the shape or size of the object

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Question

Give real three life examples of congruence transformations


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Answer

  • Sliding a paper on a tabletop (translation)

  • A number rotating in a roulette wheel (rotation)

  • Reflection on a mirror (reflection) 

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Question

Which of the below is NOT a congruent transformation?

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Answer

Dilation

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Question

What are congruence transformations rules?

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Answer

  • A congruence transformation should not change the shape or size of the object when applied to the object.

  • If one object can be moved and placed exactly over another using a sequence of congruence transformations, then the objects are congruent.

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Question

A sequence of congruent transformations will not produce congruent objects

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Answer

False

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Question

Congruence transformations arre rigid transformations

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Answer

True

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Question

Why is dilation not a congruence transformation?

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Answer

Because it changes the size of the object which breaks the rules that define a congruent transformation.

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Question

A congruence transformation that turns an object about a fixed point is...

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Answer

Rotation

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Question

Congruent figures have the

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Answer

all of the options

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Question

Name the different ways of proving congruency between triangles

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Answer

  • SSS (side, side, side): if all three sides of one triangle are the same length as the other 

  • ASA (angle, side, angle): If any two angles and one side of a triangle is equal to the other.

  • SAS (side, angle, side): If any two sides and an angle between two sides is the same as the other triangles

  • RHS: If two right-angled triangles have the hypotenuse and one side of equal length
  • Congruent transformation can be used to obtain the other shape.

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Question

Define Equilateral triangle.

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Answer

A triangle with three congruent sides is called an equilateral triangle.

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Question

Are all angles in an equilateral triangle equal?

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Answer

No

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Question

What is an isosceles triangle?

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Answer

An isosceles triangle is a triangle with two equal sides.  

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Question

"An isosceles triangle consists of two equal sides and 3 equal angles"


Is the statement above true or false? 

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Answer

False

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Question

"The base angles of an isosceles triangle are equal"


Is the statement above true or false? 

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Answer

True

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Question

"The altitude from the vertex angle of an isosceles triangle bisects its base and the vertex angle "


Is the statement above true or false? 

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Answer

True

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Question

"The altitude drawn from the apex angle divides the isosceles triangle into two similar triangles "


Is the statement above true or false? 

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Answer

False

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Question

Given the isosceles triangle ABC, how do you write the notation for the statement:


"The angles opposite the equal sides of an isosceles triangle are equal"

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Answer

Angle A = Angle B

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Question

Given the isosceles triangle ABC, how do you write the notation for the statement:


"The sides opposite the equal angles of an isosceles triangle are equal"

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Answer

AC = BC

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Question

What does the SAS Congruence rule tell us?

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Answer

If two sides and an included angle of one triangle is equal to the two sides and included angle of the second triangle, then the two triangles are said to be congruent. 

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Question

What does the ASA Congruence rule tell us?

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Answer

If two angles and an included side between the angles of one triangle are equal to the corresponding two angles and included side between the angles of the second triangle, then the two triangles are said to be congruent.

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Question

What are the three types of isosceles triangles?

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Answer

  1. Isosceles Acute
  2. Isosceles Right
  3. Isosceles Obtuse

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Question

What is the altitude of a triangle?

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Answer

An altitude is a line that passes through the vertex of a triangle that is perpendicular to the opposite side.  

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Say we have an isosceles triangle ABC. If the altitude bisects the base of the triangle at point P, what is the relationship between points A, B and P?

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Answer

AP = BP

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What is the formula for the perimeter of an isosceles triangle given the sides a and base b? 

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Answer

P = 2a + b

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Say we have an isosceles triangle ABC. If altitude to the base of an isosceles triangle bisects the vertex angle at point P, what is the relationship between angles ACP and BCP? 

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Answer

Angle ACP = Angle BCP

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Question

For an isosceles triangle ABC, the altitude to the base bisects the base at point P. Two triangles, ACP and BCP, are formed due to this bisection. What is the relationship between these two triangles?



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Answer

Triangle ACP is congruent to Triangle BCP

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Question

State the differences in the length of the sides and measures of the angles between the isosceles triangle and equilateral triangle.

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Answer

  1. An isosceles triangle has 2 equal sides while an equilateral triangle has 3 equal sides
  2. An isosceles triangle has 2 equal angles while an equilateral triangle has 3 equal angles

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Question

State the differences in the length of the sides and measures of the angles between the isosceles triangle and scalene triangle.

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Answer

  1. An isosceles triangle has 2 equal sides while a scalene  triangle has 3 unequal sides
  2. An isosceles triangle has 2 equal angles while a scalene  triangle has 3 unequal angles

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Question

What are congruent triangles?

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Answer

Congruent triangles are triangles with equal sides and angles. 

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Question

What do you do when you are unable to determine if a pair of triangles are congruent?

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Answer

 In this case, you will have to prove that another pair of triangles are congruent and then use the information gotten about its congruent sides and angles to prove that the first pair is congruent.  

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Question

What is a geometric use of congruent triangles?

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Answer

Finding distances between points.

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Question

There is a story about one of Napoleon's generals using congruent triangles. What did he use them to determine?

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Answer

He used them to determine the distance from one bank of a river to another.

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How did Napoleon's general supposedly use congruent triangles?

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Answer

The legend goes that the officer would stand at the bank of the river, and look down the brim of his cap until it was lined up with the opposite bank. Then, keeping his head at this level he would turn away from the river. Now, all he had to do was make a measurement from where he was standing to where the brim of his cap had lined up, and this would tell him the width of the river.

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Question

In what area of engineering are congruent triangles used frequently?

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Answer

They are used frequently in civil engineering and construction.

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Why are congruent triangles used so often in construction?

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Answer

Triangles display great attributes of strength and stability and congruent triangles provide an easy way to have designs aline neatly.

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Question

 What is the perimeter of an equilateral triangle if the measure of one of its sides is 4cm?

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Answer

12cm

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Question

Calculate the semiperimeter for an equilateral triangle if its perimeter is 24cm.

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Answer

Semiperimeter is 12cm.

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Question

Which of the following statement is correct?

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Answer

Equilateral triangle is a isosceles triangle

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Question

How would you define Congruent Triangles?

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Answer

One way to define Congruent Triangles: 

Congruent Triangles are ones of the same shape and size and can be positioned differently in space.

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Question

How many triangles need to be given to talk about congruence?

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Answer

Two or more

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Question

In what cases are two triangles considered congruent?

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Answer

When they have the same shape

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Question

Two equilateral triangles are given. Both have a side of 6cm in length. Are these triangles congruent?

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Answer

Yes

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Question

What are considered Non-Congruent triangles?

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Answer

Triangles that differ in size and/or shape are considered non-congruent

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Question

If two triangles differ in size but have the same shape, does this mean they are congruent?

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Answer

Yes

Show question

Question

If two triangles are positioned differently in space (have a gap between them, rotated relative to another) but have the same shape and size, does this mean they are congruent?

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Answer

Yes

Show question

Question

Do you always need to find out the length of every side and the value of every angle to evaluate whether two triangles are congruent?

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Answer

No, there are some handy alternatives to measuring every side and angle. 

You can use one of the five theorems that help evaluate triangle congruence.

Show question

Question

Which of these are not examples of congruent triangles?

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Answer

Two equilateral triangles with equal sides

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