Overview of the unit
TRI 01-Classify triangles by sides and angles
These are the different types of ways you can classify a triangle. A triangle is classified by its sides and angles. Here are some examples below. Also, when naming a triangle, you state the angle classification first, and then the side classification.
Angles |
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Tri 02-Fill in the missing statements/reasons in a 2 column proof of the triangle angle sum theorem
The triangle angle sum theorem proof is trying to prove that all of the interior angles in a triangle will equal 180 degrees. Here is the proof below.
Tri 03-Given two angles of a triangle, find the third
Given this triangle
Create an algebraic expression for the three angles in a triangle, solve for x and find the measure of each angle.
Since we proved that the sum of a triangle's interior angles would add up to 180 degrees we could create the equation:
m∠a + m∠b + m∠c = 180 degrees
We already know from the image that m∠a = 30 degrees and m∠b = 37 degrees so we can plug that into the equation using substitution, so
30 + 37 + m∠c = 180
Then, just solve the equation to find the measure of angle c.
67 + m∠c =180
-67 -67
m∠c = 113 degrees
So the missing angle, m∠c, is 113 degrees.
Since we proved that the sum of a triangle's interior angles would add up to 180 degrees we could create the equation:
m∠a + m∠b + m∠c = 180 degrees
We already know from the image that m∠a = 30 degrees and m∠b = 37 degrees so we can plug that into the equation using substitution, so
30 + 37 + m∠c = 180
Then, just solve the equation to find the measure of angle c.
67 + m∠c =180
-67 -67
m∠c = 113 degrees
So the missing angle, m∠c, is 113 degrees.
Tri 04-STATE THE COROLLARIes to the triangle angle sum theorem
Here are some corollaries to the triangle angle sum theorem
a. In any triangle, no more than one angle can be right.
b. In a right triangle, the two non-right angles are complementary.
c. If two angles of one triangle are congruent to two angles of another, then the third angles are also congruent.
d. In any triangle, no more than one angle can be obtuse.
a. In any triangle, no more than one angle can be right.
b. In a right triangle, the two non-right angles are complementary.
c. If two angles of one triangle are congruent to two angles of another, then the third angles are also congruent.
d. In any triangle, no more than one angle can be obtuse.
tri 05-state and apply the remote interior angles theorem (aka exterior angle theorem)
The theorem states,
The measure of an exterior angle is equal to the sum of the two remote interior angles.
Now, let's apply it
The measure of an exterior angle is equal to the sum of the two remote interior angles.
Now, let's apply it
Based on the theorem, we know that the interior angles (∠1 and ∠2) are equal to the remote exterior angle (∠4). So we can right an equation:
m∠1 + m∠2 = m∠4 If we say that m∠1 = 40 and m∠2 = 35, we can solve the equation Now, substitute 40 + 35 = m∠4 75 = m∠4 m∠4 =75 From this equation, we know that the m∠4 is equal to 75 degrees. |
tri 06-use the triangle inequality theorem
Determine if 3 given sides form a triangle. Also be able to determine the range of values for a third side, given the other two.
In this theorem, the side lengths for a triangle are only possible if the two smallest side lengths are greater than the biggest one.
The expression for this is |a-b| < x < a + b
A triangle with side lengths 5, 23, and 15 would NOT work because when we plug it in,
|15-5| < 23 < 15 +5
10 < 23 < 20, This does NOT work
but if we had the numbers 10, 23, and 15 it WOULD work because
|15-10| < 23 < 15 + 10
5 < 23 < 25, This DOES work
We can also use this to determine the range of values for the third side, given two.
If we had the numbers 7 and 9 we know that the range of values for the third side would have to be
2 < x < 16
because |9-7| < x < 9+7 which is from the inequality |a-b| < x < a + b
In this theorem, the side lengths for a triangle are only possible if the two smallest side lengths are greater than the biggest one.
The expression for this is |a-b| < x < a + b
A triangle with side lengths 5, 23, and 15 would NOT work because when we plug it in,
|15-5| < 23 < 15 +5
10 < 23 < 20, This does NOT work
but if we had the numbers 10, 23, and 15 it WOULD work because
|15-10| < 23 < 15 + 10
5 < 23 < 25, This DOES work
We can also use this to determine the range of values for the third side, given two.
If we had the numbers 7 and 9 we know that the range of values for the third side would have to be
2 < x < 16
because |9-7| < x < 9+7 which is from the inequality |a-b| < x < a + b
tri 07-Given two angles in a scalene triangle, list the triangle's side lengths in descending order and vice versa
First we need to know that the biggest side is opposite the biggest angle, the smallest side is opposite the smallest angle, and the middle side is opposite the middle angle. When listing in ascending order, the side lengths organize from smallest measure to largest measure and in descending order it is organized from largest measure to smallest measure.
Here are some examples of how to apply it.
Here are some examples of how to apply it.
*CONSTRUCTIONS ARE BELOW ALL OF THE SKILLS
TRI 08-DEFINE POLYGON CONGRUENCE
Define polygon: A shape is a polygon iff
-an enclosed figure (all sides terminate at a vertex)
-many sides (greater than 2)
-all sides are straight
-is coplanar
DEFINITION OF POLYGON CONGRUENCE: 2 or more polygons are congruent iff all of the corresponding sides are congruent and all of the corresponding angles are congruent.
-an enclosed figure (all sides terminate at a vertex)
-many sides (greater than 2)
-all sides are straight
-is coplanar
DEFINITION OF POLYGON CONGRUENCE: 2 or more polygons are congruent iff all of the corresponding sides are congruent and all of the corresponding angles are congruent.
tri 09-Given two polygons, determine if they are congruent by the definition of polygon congruence.
tri 10-given two congruent polygons, write polygon congruence statements and identify congruent corresponding parts
In congruence statements, order matters, so you must match up the letters with their placement on the triangle. It does NOT have to be in alphabetical order, as you can see in the picture above. Using the side length and angle congruence marks, you can use the as a guideline to match up the letters.
We can then use this to find out the corresponding parts because since they are in order we know information such as, angle X is congruent to angle T
We can then use this to find out the corresponding parts because since they are in order we know information such as, angle X is congruent to angle T
tri 11-accurately express the definition of congruence, orally or in writing, including references to both transformations and corresponding parts.
Definitions
-Using corresponding parts: Two triangles are congruent if all corresponding sides and all corresponding angles are congruent.
-Using transformations: Two triangles are congruent iff one can be mapped onto the other using a series of rigid transformation.
4 transformations in geometry are
-translation (a slide)
-rotation (Rotating around the origin by 180 degrees can be found by (~x, ~y)
-reflection (flipped over x axis is (x,~y), flipped over y axis is (~x, y), and (~x, ~y) is flipped over both axes
-dilation (this one is not a rigid transformation like the others because the pre image is not congruent to the new image)
Click HERE to use this website link to practice doing transformations
-Using corresponding parts: Two triangles are congruent if all corresponding sides and all corresponding angles are congruent.
-Using transformations: Two triangles are congruent iff one can be mapped onto the other using a series of rigid transformation.
4 transformations in geometry are
-translation (a slide)
-rotation (Rotating around the origin by 180 degrees can be found by (~x, ~y)
-reflection (flipped over x axis is (x,~y), flipped over y axis is (~x, y), and (~x, ~y) is flipped over both axes
-dilation (this one is not a rigid transformation like the others because the pre image is not congruent to the new image)
Click HERE to use this website link to practice doing transformations
tri 12-given one or more triangles with congruence marks on sides or angles, students will be able to correctly identify which parts are congruent and which are not known to be so
Know the congruence marks for sides and angles. Use this video for help to determine congruence.
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tri 13-when shown a triangle with three components indicated by congruence marks, student will accurately "name" the relationship among congruent elements as aaa, aas, asa, saa, ass, sas, ssa, or sss
We know that two triangles are congruent iff all corresponding angles and all corresponding sides are congruent, but what if there is a shorter way to prove congruence? When we test it, one angle or two angles is not enough because the sides could be different, and one or two sides is not enough because the angles could be different. However, when we test 3 sides, we find that it IS enough information to prove congruence, while 3 angles is NOT enough.
Look at this table below to see all of the different combos you can have in a triangle. The ones in green do contain enough information to prove congruence while the ones in the red do not.
Look at this table below to see all of the different combos you can have in a triangle. The ones in green do contain enough information to prove congruence while the ones in the red do not.
tri 14-when faced with the potentially ambiguously named triangles (aas/ssa and aas/ssa) students will be able to, with 100% certainty, choose the one that is alphabetically first.
As you can see by the table above, you might notice that AAS and SAA and ASS and SSA appear to be the same type of triangle congruence, just in reverse order. That is because they are. Since those two pairs are exactly the same, you would choose the one in alphabetical order, which are the ones that start with an A. This basically means that you would always use AAS, instead of SAA, and you would always use ASS, instead of SSA.
tri 15-Students are able to determine which of the triangle congruence theorems (aas, asa, sas, sss) are sufficient to prove triangle congruence, and which are not (aaa, ass)
These are the pairs that DO prove congruence:
AAS, ASA, SAS, SSS
These are the pairs that DO NOT prove congruence:
AAA, ASS
These are the pairs we will not use:
SSA, SAA
AAS, ASA, SAS, SSS
These are the pairs that DO NOT prove congruence:
AAA, ASS
These are the pairs we will not use:
SSA, SAA
tri 16-When given a pair of triangles, with appropriate congruence marks, students will be able to determine whether or not there is sufficient and appropriate information to determine triangle congruence using the triangle congruence theorems.
Using the Mario Kart Analogy, you can pretend that you are driving out the "race track", which is the triangle. If you see an angle congruent mark, you pick up an A. If you see a side congruent mark, you pick up an S. If you go past one angle or side that does not have ANY marks, then it slows your car down, but you can keep going. If you go past TWO angles or sides that do not have any marks, then it is game over. This will help you determine what type of triangle it is, and if you can prove its congruence.
Let's do an example:
Let's do an example:
tri 17-when claiming a triangle pair is congruent, students will correctly express the triangle congruence statement.
As we learned in TRI 10, writing polygon congruence statements must be in the order they have been given. This is the same for triangle congruence statements. Let's practice.
tri 18-given two triangles, with explicitly stated three appropriate congruent facts, sufficient to be able to determine the triangles are congruent, students will write a simple two column proof to show that two triangles are congruent, with 100% accuracy.
In this type of proof, there are only 4 steps. The first 3 are writing the 3 given portions and putting their congruence marks on the triangle, and the last one is determining what type of triangle congruence theorem it is (AAS, ASA, SAS, SSS). Here is a worksheet to practice these proofs and a follow up key to check your answers.
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tri 19-be able to distinguish Ass from rhl, and use rhl to determine if two triangles are congruent.
Rather than classify a triangle like the one at right as an ASS triangle, which does not contain enough information to prove congruence, you can use RHL. RHL stands for right hypotenuse leg, and you can figure it out based on previous knowledge of the pythagorean theorem. If you have a right angle, and a congruent side and hypotenuse, then you know that the other leg will also be congruent. This means that now you have an SSS triangle, which does prove congruence, so therefore, RHL proves congruence also.
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tri 20-write simple triangle congruence proofs in a flow proof format
Flow proofs are simply proofs expressed in a more visual way, rather than the standard columns. You can take really any proof you want and turn one into a flow proof. Here is an example.
write two column or flow proofs for more complicated triangle congruence (including reflexive, symmetric, parallel lines, vertical angles, cpctc (corresponding parts of congruent triangles are congruent), etc.)
There are many different types of two column or flow proofs you can write so here is a worksheet of many you can try out.
tri 50-given a triangle and designated parts, construct significant segments
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tri 51-Given a triangle construct the centroid. describe its significance
Forming the centroid finds the center of gravity of a triangle. 3 medians
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tri 52-given a triangle construct the incenter. demonstrate its significance
The center of the circle is equidistant from all three sides. (inside triangle) made by 3 angle bisectors
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tri 53-given a triangle construct the circumcenter. Demonstrate its significance
The center of the circle contains all 3 vertices or is equidistant from all 3 vertices. 3 perpendicular bisectors
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tri 54-GIVEN A TRIANGLE CONSTRUCT THE orthocenter. DEMONSTRATE ITS SIGNIFICANCE
The orthocenter is the point of concurrency produced by the construction of 3 altitudes. The orthocenter contributes to the idea of the Euler line.
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