Math-Essentials Lesson 6-2jefflongnuames.weebly.com/uploads/5/5/8/6/55860113/math...Naming Triangles...
Transcript of Math-Essentials Lesson 6-2jefflongnuames.weebly.com/uploads/5/5/8/6/55860113/math...Naming Triangles...
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Triangle Congruence
Math-Essentials
Lesson 6-2
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Naming TrianglesTriangles are named using a small triangle symbol and the three vertices of the triangles.
The order of the vertices does not matter for NAMING a triangle
Examples
A C
BΞπ΄π΅πΆΞπ΄πΆπ΅Ξπ΅π΄πΆΞπ΅πΆπ΄ΞπΆπ΄π΅
ΞπΆπ΅π΄
X
Z
Y
Ξπππ
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Your TurnGive all six names of the triangles
P R
Q
Ξπππ Ξππ πΞπ ππΞπ ππΞπππ
Ξππ π
D
F
E
1) 2)Ξπ·πΈπΉΞπ·πΉπΈΞπΈπ·πΉΞπΈπΉπ·ΞπΉπ·πΈ
ΞπΉπΈπ·
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Correspondence
In triangles, there are two types of correspondence that we consider:β’ Corresponding angles
β’ Corresponding sides
Corresponding Angles of Triangles: an angle in one triangle that has the same position (relative to its sides) as an angle in another triangle (relative to its sides).
Corresponding Sides of Triangles: a side in one triangle that has the same position (relative to its angles) as a side in another triangle (relative to its angles).
β π΄ corresponds to β π· since they are opposite the longest side of their triangles
π΅πΆ corresponds to πΈπΉ since they are opposite the largest angle of their triangles
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Your Turn:
1) What angle does β π΄ correspond to?
2) What angle does β π correspond to?
3) What side does ππ correspond to?
4) What side does π΄πΆ correspond to?
β π
β πΆ
πΆπ΅
ππ
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Congruence
β’ Anglesβ’ two angles are congruent if they have the same measure (degrees)β’ β π΄ β β π΅ if πβ π΄ = πβ π΅
β’ Segments (sides)β’ two line segments are congruent if they have the same lengthβ’ π΄π΅ β πΆπ· if π΄π΅ = πΆπ·
β’ Trianglesβ’ two triangles are congruent if each angle in one triangle is congruent to its
corresponding angle in the other triangle AND if each side in one triangle is congruent to its corresponding side in the other polygon.
β’ In short we say βcorresponding parts of congruent triangles are congruentβ or βCPCTCβ
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Congruence Statements
β’ When naming a triangle, the order of the vertices is not important
β’ But when stating congruence, the order is importantβ’ The vertices of must be put in order so that the corresponding parts in the
names of the triangles match the corresponding parts in the triangles themselves
β’ For Example, Ξπ΄π΅πΆ β Ξπππ becauseβ¦ β’ β π΄ corresponds to β π
β π΅ corresponds to β πβ πΆ corresponds to β π
β’ π΄π΅ corresponds to πππ΅πΆ corresponds to πππΆπ΄ corresponds to ππ
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Your Turnβ’ Express the congruence of the following pairs of triangles
1) 2)
Ξπ·πΈπΉ β Ξππ π Ξπ ππ β ΞπΊπΉπ»
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Congruence Conditions
Why are Ξπ ππ and Ξπππ congruent? (That is, how do we prove it?)
β’ The corresponding parts are congruent (CPCTC)
β’ This is just the definition of congruence
Are there other ways we can know triangles are congruent?...
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Background Vocab
β’ Included side: If two angles in a triangle are given, the included side is the side that is between the two angles or side that both of the angles have in common.
β’ π π is the included side of β π and β π
β’ Included angle: If two sides of a triangle are given, the included angle is the angle formed by those two sides.
β’ β π is the included angle of π π and ππ
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D
F
E
Your Turn
1) β π· is the included angle of which two sides?
2) What is the included angle of sides π·πΉ and πΈπΉ?
3) π·πΉ is the included side of which two angles?
4) What is the included side of β π· and β πΈ
π·πΉ and π·πΈ
β πΉ
β π· and β πΉ
π·πΈ
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More Congruence Conditions
Side-Angle-Side (SAS) Congruence Axiom: if an angle in one triangle is congruent to an angle in another triangle and if the sides of the angle in the first triangle are congruent to the sides of the angle in the other triangle, then the two triangles are congruent.
β’ ππ β ππ
β’ β πππ β β π ππΉ
β’ ππ β ππΉ
β’ Therefore, Ξπππ β ΞππΉπ by SAS
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More Congruence Conditions
β’ Angle-Side-Angle (ASA) Congruence Theorem: if two angles and their included side are congruent, then the two triangles are congruent.
β’ β π΄π΅πΆ β β π·πΈπΊ
β’ π΅πΆ β πΈπΊ
β’ β π΅πΆπ΄ β β πΈπΊπ·
β’ Therefore, Ξπ΄π΅πΆ β Ξπ·πΈπΊby ASA
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More Congruence Conditions
Side-Side-Side (SSS) Congruence Theorem: if all corresponding sides of a triangle are congruent, then the triangles are congruent
β’ π΄π΅ β π·πΈ
β’ π΅πΆ β πΈπΉ
β’ πΆπ΄ β πΉπ·
β’ Therefore, Ξπ΄π΅πΆ β Ξπ·πΈπΉby SSS
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More Congruence Conditions
β’ Angle-Angle-Side (AAS) Congruency Theorem: If two corresponding angles are congruent between two triangles and a pair of corresponding sides are congruent (which are NOT the included side), then the two triangles are congruent.
β’ β πππ β β πΈπΉπ·
β’ β πππ β β πΉπ·πΈ
β’ ππ β πΉπΈ
β’ Therefore, Ξπππ β ΞπΉπ·πΈby AAS
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Your Turnβ’ Determine which congruence condition proves the congruence for each
the following pairs of triangles.
β’ Write a congruence statement for each of the following pairs of triangles.
1) 2)
3) 4)
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Your Turn
You may have notice a pattern of needing 3 parts of a triangle.
What other 3-part groups are we missing?
β’ AAA
β’ ASS (usually referred to by the more appropriate SSA)
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Angle-Angle-Angle (AAA) Condition
Letβs look at an example of AAA
β’ β π΄ β β π΄
β’ β π΄π΅πΆ β β π΄π·πΈ (why?)
β’ β π΄πΆπ΅ β β π΄πΈπ· (why?)
β’ Does this mean that Ξπ΄π΅πΆ β Ξπ΄π·πΈ? How do you know?
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Angle-Side-Side (ASS) Condition
Letβs look at an example of ASS
β’ β π΄ β β π·
β’ π΄π΅ β π·πΈ
β’ π΅πΆ β πΈπΉ
β’ Does this mean that Ξπ΄π΅πΆ β Ξπ·πΈπΉ? How do you know?
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Congruence Conditions
β’ Conditions that work (axioms)β’ SSS
β’ ASA
β’ SAS
β’ AAS
β’ Conditions that do not workβ’ AAA
β’ ASS