Common Core: High School - Geometry : Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

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All Common Core: High School - Geometry Resources

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Example Questions

Example Question #497 : High School: Geometry

What is rigid motion?

Possible Answers:

Any way of moving a figure such that the relative distance between the points/vertices of the figure stay the same and the relative position of the points/vertices of the figure stay the same

Any way of moving a figure such that the relative distance between the points/vertices of the figure stay the same but the position can differ

Any way of moving a figure such that the relative position of the points/vertices of the figure stay the same but the distance between points/vertices can differ

Any way of moving a figure

Correct answer:

Any way of moving a figure such that the relative distance between the points/vertices of the figure stay the same and the relative position of the points/vertices of the figure stay the same

Explanation:

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.

Example Question #111 : Congruence

In terms of rigid motion, how do we know when two figures are congruent to one another?

Possible Answers:

Two figures are congruent if they meet the criteria of all three of the following theorems: SAS, ASA, SSS

Two figures are congruent if there is a sequence of rigid motions that maps one figure to another

Two figures are congruent if there is a sequence of rigid motions that maps at least two vertices to another

Two figures are congruent if they meet the criteria of one of the following theorems: SAS, ASA, SSS

Correct answer:

Two figures are congruent if there is a sequence of rigid motions that maps one figure to another

Explanation:

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

 

         

Example Question #499 : High School: Geometry

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)

Possible Answers:

Reflection, rotation

Rotation

Translation

Translation, rotation, reflection

Correct answer:

Translation, rotation, reflection

Explanation:

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.

 

 

Example Question #2 : Explain How The Criteria For Triangle Congruence (Asa, Sas, And Sss) Follow From The Definition Of Congruence In Terms Of Rigid Motions.

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.

Possible Answers:

False

True

Correct answer:

True

Explanation:

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.

 

Example Question #112 : Congruence

Triangles that share a side and follow the SSS criteria for congruence follow which of the following rigid motions?

Possible Answers:

Translation

None of the answer choices are correct

Reflection

Rotation

Correct answer:

Reflection

Explanation:

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.

 

Example Question #4 : Explain How The Criteria For Triangle Congruence (Asa, Sas, And Sss) Follow From The Definition Of Congruence In Terms Of Rigid Motions.

True or False: The following triangles are congruent by two different methods:

Q4 2

 

Possible Answers:

False

True

Correct answer:

True

Explanation:

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.

Example Question #5 : Explain How The Criteria For Triangle Congruence (Asa, Sas, And Sss) Follow From The Definition Of Congruence In Terms Of Rigid Motions.

Tell why the following triangles are congruent both by rigid motions and one of the three triangle congruence theorems.

Screen shot 2020 08 13 at 8.59.21 am

Possible Answers:

SSS, rotation

SSS, reflection

SAS, reflection

SAS, translation

Correct answer:

SSS, reflection

Explanation:

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.

Example Question #1 : Explain How The Criteria For Triangle Congruence (Asa, Sas, And Sss) Follow From The Definition Of Congruence In Terms Of Rigid Motions.

Through which rigid motion are the following triangles related by?

Screen shot 2020 08 20 at 10.31.35 am

Possible Answers:

Translation

None of the choices are correct

Reflection

Rotation

Correct answer:

Reflection

Explanation:

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.

Screen shot 2020 08 20 at 10.32.07 am

Example Question #467 : High School: Geometry

Give an informal proof that proves the following two triangles are congruent by the SAS Theorem and by a series of rigid motions.

Screen shot 2020 08 20 at 11.22.02 am

Possible Answers:

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 ASA theorem. We are able to map  to  by reflecting . 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  and so these two triangles are congruent by the ASA theorem. We are able to map  to  by translating . 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   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.

Correct answer:

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.

Explanation:

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.

Example Question #1 : Explain How The Criteria For Triangle Congruence (Asa, Sas, And Sss) Follow From The Definition Of Congruence In Terms Of Rigid Motions.

The following two triangles are congruent by the ASA Theorem.  What are the series of rigid motions that map them to one another?

Screen shot 2020 08 20 at 11.23.37 am

Possible Answers:

Translation, rotation

Reflection, translation

Rotation, reflection

Translation

Correct answer:

Rotation, reflection

Explanation:

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 .

Screen shot 2020 08 20 at 11.24.12 am

Now we can reflect the triangle  across  to map  to  to  and  to .

Screen shot 2020 08 20 at 11.24.19 am

So the order of rigid motions is rotation, reflection.

All Common Core: High School - Geometry Resources

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