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### Course: High school geometry>Unit 3

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.
(1 vote)
• What's the difference between similar and congruent?
(1 vote)
• 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.
• 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?
(1 vote)
• 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?
• What is the formula for a triangle
• 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.
• At he says that an angle is defined by two rays, can’t an angle be defined by a line/line segment as well?
(1 vote)
• What is the => thing that he drew?
• I don't really think that's an actual math symbol (Or at least in this context), but rather just an indicator saying that the left side of the arrow concludes right side of the arrow.
(1 vote)
• 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."