Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. - Common Core: High School - Geometry
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What is rigid motion?
What is rigid motion?
Rigid motion follows these criteria because the motion is rigid meaning that everything that is moving stays the same except for the location of the entire figure. There are three common types of rigid motion; translation, reflection, and rotation.
Rigid motion follows these criteria because the motion is rigid meaning that everything that is moving stays the same except for the location of the entire figure. There are three common types of rigid motion; translation, reflection, and rotation.
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In terms of rigid motion, how do we know when two figures are congruent to one another?
In terms of rigid motion, how do we know when two figures are congruent to one another?
This is the correct definition in terms of rigid motions. Some of the other options are correct definitions for congruence but do not mention the criteria of there being rigid motion between the two figures. An example of this is that 
and 
are congruent because they are a reflection of one another. Their vertices that map to each other are
This is the correct definition in terms of rigid motions. Some of the other options are correct definitions for congruence but do not mention the criteria of there being rigid motion between the two figures. An example of this is that 
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True or False: The following triangles are congruent by two different methods:
True or False: The following triangles are congruent by two different methods:
Let’s first begin by showing that these two triangles are congruent through a series of rigid motions. Let’s use our point of reference be 
and 
since we know that these angles are congruent through the information given in the picture. We are able to map 
to 
by reflecting 
along the line 
. So these two triangles are congruent.
Now we will show that these two triangles are congruent through another theorem. We see that there are two pairs of corresponding congruent angles, 
, and the angles’ included sides are congruent as well, 
. The ASA Theorem states that if two triangles share two pairs of corresponding congruent angles and their included sides are congruent, then these two triangles are congruent. So by ASA, these triangles are congruent.
Let’s first begin by showing that these two triangles are congruent through a series of rigid motions. Let’s use our point of reference be 
Now we will show that these two triangles are congruent through another theorem. We see that there are two pairs of corresponding congruent angles, 
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The following two triangles are congruent by the SAS Theorem. What are the series of rigid motions that map them to one another? (Figures not to scale)
The following two triangles are congruent by the SAS Theorem. What are the series of rigid motions that map them to one another? (Figures not to scale)
First, we need to establish a vector that maps at least one pair of vertices. We will use 
to establish a translation between the two figures. This also maps 
to 
.
Now they share a vertex and we are able to rotate them together mapping 
to 
and 
to 
.
Now we can reflect across 
mapping 
to 
, 
to 
, and 
to 
.
So the order of the series of rigid motions is translation, rotation, reflection.
First, we need to establish a vector that maps at least one pair of vertices. We will use 
Now they share a vertex and we are able to rotate them together mapping 
Now we can reflect across 
So the order of the series of rigid motions is translation, rotation, reflection.
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The following two triangles are congruent by the ASA Theorem. What are the series of rigid motions that map them to one another?
The following two triangles are congruent by the ASA Theorem. What are the series of rigid motions that map them to one another?
First, the two triangles 
and 
share a vertex, so we know that 
maps to 
by the reflective property. Knowing this, we are able to rotate 
to match the congruent sides 
and 
. This maps 
to 
. We can also note that 
maps to 
.
Now we can reflect the triangle 
across 
to map 
to 
, 
to 
and 
to 
.
So the order of rigid motions is rotation, reflection.
First, the two triangles 
Now we can reflect the triangle 
So the order of rigid motions is rotation, reflection.
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Give an informal proof that proves the following two triangles are congruent by the SAS Theorem and by a series of rigid motions.
Give an informal proof that proves the following two triangles are congruent by the SAS Theorem and by a series of rigid motions.
The SAS Theorem states that if two triangles share two pairs of corresponding congruent sides are congruent and their included angle is also congruent, then these two triangles are congruent. We are given that 
and so these two triangles are congruent by the SAS theorem. We are able to map 
to 
by rotating 
180 degrees clockwise. So these two triangles are congruent by rigid motion.
The SAS Theorem states that if two triangles share two pairs of corresponding congruent sides are congruent and their included angle is also congruent, then these two triangles are congruent. We are given that 
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True or False: If two triangles are congruent through SAS Theorem and share a vertex, they will follow the rigid motions of rotation and reflection.
True or False: If two triangles are congruent through SAS Theorem and share a vertex, they will follow the rigid motions of rotation and reflection.
Consider the triangles 
and 
, where 
. We are able to rotate them together mapping 
to 
and 
to 
.
Now we can reflect across 
mapping 
ro 
, and 
to 
.
Now we are left with the two congruent triangles lying on top of one another, proving that the rigid motions that map these two triangles to one another are rotation and reflection.
Consider the triangles 
Now we can reflect across 
Now we are left with the two congruent triangles lying on top of one another, proving that the rigid motions that map these two triangles to one another are rotation and reflection.
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Triangles that share a side and follow the SSS criteria for congruence follow which of the following rigid motions?
Triangles that share a side and follow the SSS criteria for congruence follow which of the following rigid motions?
Consider the following triangles, 
and 
. They share the side 
. If we reflect triangle 
across 
, we match up all congruent sides, mapping them to one another and mapping 
to 
, 
to 
, and 
to 
, proving these two triangles congruent.
Consider the following triangles, 
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Through which rigid motion are the following triangles related by?
Through which rigid motion are the following triangles related by?
This becomes more clear with the orange line between the two triangles. If flipped over this orange line, the two figures would match up their corresponding congruent parts creating the same triangle.
This becomes more clear with the orange line between the two triangles. If flipped over this orange line, the two figures would match up their corresponding congruent parts creating the same triangle.
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Tell why the following triangles are congruent both by rigid motions and one of the three triangle congruence theorems.
Tell why the following triangles are congruent both by rigid motions and one of the three triangle congruence theorems.
We can see that 
. We know that 
by reflexive property. So by the SSS Theorem, these two triangles are congruent. We can also reflect triangle 
across line 
to map the remaining angles to one another; 
to 
. So these triangles are proven congruent through reflection as well.
We can see that 
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