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### Course: Geometry (FL B.E.S.T.)>Unit 4

Lesson 1: Transformations & congruence

# Angle congruence equivalent to having same measure

Two angles are congruent if and only if they have the same measure.

## Want to join the conversation?

• Hello, is there a difference between Congruent and Equivalent?
• Congruent is a term specific to geometry. Two figures are congruent when you can map one perfectly onto the other by reflection, rotating, and translating it without distortion.

Equivalent is a more general term. In general, it means 'the same as, in all relevant ways.' In the context of the video title 'Angle congruence equivalent to having same measure', it means that the sentence "These angles are congruent" is true exactly when "These angles have the same measure" is true.

That is, if these angles are congruent, then they have the same measure. And if they are not congruent, then they have different measures.
• At , what's "prime"? When would we see that? And how would we write that?
• This is a second meaning of prime in math, like in the transformers movies (Optimus Prime). It is an indication of a point that has been moved by one of the transformation (translation, rotation, reflection, or dilation) and is represented as A' B' or C'. So the original point A is moved to A', the original point B is moved to B', and the original point C is moved to C'.
You could even do two transformations on a point and end up with A double prime A".
• so does congruent (≅) mean both equal (=) and similar (~)
• congruent is things that are equal length, size etc. Two shapes that have the same proprtions (angles, sides etc) but are not necesarily the same size ( basically one is a scaled up version of the other)are similar. So an angle wih the sides 3, 4, and 5 will be similar to one with the sides of 6,, and 10, but not congruent.
• in the previous video, rigid transformation is defined as transformations that preserve distance between points.

in this video, it adds "angle measures". Is it possible to prove that angle measures preservation is a consequence of "distance preservation"?
(1 vote)
• Let's say we have some shape 𝑆 that undergoes a rigid transformation, resulting in the image 𝑆'.

A rigid transformation preserves distances, so for any three distinct points 𝐴, 𝐵, 𝐶 on 𝑆 and their corresponding points 𝐴ʹ, 𝐵ʹ, 𝐶ʹ on 𝑆ʹ
we have 𝐴𝐵 = 𝐴ʹ𝐵ʹ, 𝐴𝐶 = 𝐴ʹ𝐶ʹ and 𝐵𝐶 = 𝐵ʹ𝐶ʹ.

By side-side-side congruence, we then have △𝐴𝐵𝐶 ≊ △𝐴ʹ𝐵ʹ𝐶ʹ.

Because the two triangles are congruent their corresponding angles are also congruent and thereby equal in measure:
𝑚∠𝐶𝐴𝐵 = 𝑚∠𝐶ʹ𝐴ʹ𝐵ʹ
𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐴ʹ𝐵ʹ𝐶ʹ
𝑚∠𝐵𝐶𝐴 = 𝑚∠𝐵ʹ𝐶ʹ𝐴ʹ

Thus we have shown that preservation of distances implies preservation of angle measures.

Specifically, 𝑚∠𝑉 = 𝑚∠𝑉' for any vertex 𝑉 on 𝑆 and its corresponding vertex 𝑉' on 𝑆'.
(1 vote)
• What is the formula for a triangle
• Well there at a lot of formulas for a triangle
You have area, perimiter, missing angles, missing triangle lengths. Just google the one you want
• What's the difference between similar and congruent?
• For congruent shapes, they must be exactly the same shape.

For similar shapes, another shape can be obtained by enlarging or reducing the size of the shape.

For example, all circles are similar, since you can enlarge or reduce the size of the circle to obtain another circle.

Note that for shapes that are congruent, they are also similar.
• At he says that an angle is defined by two rays, can’t an angle be defined by a line/line segment as well?
• How would I solve this word problem: The perimeter of a square is 4 units greater than the combined perimeter of two congruent equilateral triangles. If the side length of the square is 10 units, what is the side length of the triangles?
• First, you would have to find the perimeter of a square. Since each side length is 10, the perimeter is 10+10+10+10, or 40.

It says that the perimeter of the square is 4 units greater than the combined perimeter of the congruent equilateral triangles.

So, 40-4, or 36, is the total perimeter of both of the congruent equilateral triangles. Divide this by six to get each side of the triangle, since there are 6 equal sides in 2 congruent equilateral triangles.

36/6 is 6, so the side length of each side of the triangles is 6 units.

Does this make sense?
(1 vote)
• Are ∠ABC and ∠DEF congruent even if the lengths of AB and DE are not equal, if they have the same measure?

Or is it ambiguous and depends on the problem?
• The lengths do not have to be equal. You can extend or shorten the lengths of the sides without affecting the angle measure at all. Just draw an angle on a piece of paper and measure the angle with a protractor. Now extend the angle. No matter how much you extend it, the angle will be exactly the same.
• According to Wikipedia, "In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.[1][self-published source][2][3]

The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a proper rigid transformation, or rototranslation.[citation needed] Any proper rigid transformation can be decomposed into a rotation followed by a translation, while any improper rigid transformation can be decomposed into an improper rotation followed by a translation, or into a sequence of reflections.

Any object will keep the same shape and size after a proper rigid transformation.

All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a mathematical group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of proper rigid transformations is called special Euclidean group, denoted SE(n).

In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to represent the linear and angular displacement of rigid bodies. According to Chasles' theorem, every rigid transformation can be expressed as a screw displacement."
• Thats a lot of writing. I am not going to read that all
(1 vote)

## Video transcript

- [Instructor] What we're going to do in this video is demonstrate that angles are congruent if and only if they have the same measure, and for our definition of congruence, we will use the rigid transformation definition, which tells us two figures are congruent if and only if there exists a series of rigid transformations which will map one figure onto the other. And then, what are rigid transformations? Those are transformations that preserve distance between points and angle measures. So, let's get to it. So, let's start with two angles that are congruent, and I'm going to show that they have the same measure. I'm going to demonstrate that, so they start congruent, so these two angles are congruent to each other. Now, this means by the rigid transformation definition of congruence, there is a series of rigid transformations, transformations that map angle ABC onto angle, I'll do it here, onto angle DEF. By definition, by definition of rigid transformations, they preserve angle measure, preserve angle measure. So, if you're able to map the left angle onto the right angle, and in doing so, you did it with transformations that preserved angle measure, they must now have the same angle measure. We now know that the measure of angle ABC is equal to the measure of angle DEF. So, we've demonstrated this green statement the first way, that if things are congruent, they will have the same measure. Now, let's prove it the other way around. So now, let's start with the idea that measure of angle ABC is equal to the measure of angle DEF, and to demonstrate that these are going to be congruent, we just have to show that there's always a series of rigid transformations that will map angle ABC onto angle DEF, and to help us there, let's just visualize these angles, so, draw this really fast, angle ABC, and angle is defined by two rays that start at a point. That point is the vertex, so that's ABC, and then let me draw angle DEF. So, that might look something like this, DEF, and what we will now do is let's do our first rigid transformation. Let's translate, translate angle ABC so that B mapped to point E, and if we did that, so we're gonna translate it like that, then ABC is going to look something like, ABC is gonna look something like this. It's going to look something like this. B is now mapped onto E. This would be where A would get mapped to. This would where C would get mapped to. Sometimes you might see a notation A prime, C prime, and this is where B would get mapped to, and then the next thing I would do is I would rotate angle ABC about its vertex, about B, so that ray BC, ray BC, coincides, coincides with ray EF. Now, you're just gonna rotate the whole angle that way so that now, ray BC coincides with ray EF. Well, you might be saying, "Hey, C doesn't necessarily have "to sit on F 'cause they might be different distances "from their vertices," but that's all right. The ray can be defined by any point that sits on that ray, so now, if you do this rotation, and ray BC coincides with ray EF, now those two rays would be equivalent because measure of angle ABC is equal to the measure of angle DEF. That will also tell us that ray BA, ray BA now coincides, coincides with ray ED, and just like that, I've given you a series of rigid transformations that will always work. If you translate so that the vertices are mapped onto each other and then you rotate it so that the bottom ray of one angle coincides with the bottom ray of the other angle, then you could say the top ray of the two angles will now coincide because the angles have the same measure, and because of that, the angles now completely coincide, and so we know that angle ABC is congruent to angle DEF, and we're now done. We've proven both sides of this statement. If they're congruent, they have the same measure. If they have the same measure, then they are congruent.