Post on 05-Feb-2022
Lesson 5-2 Triangle Congruence
Math-2
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΄πΆ
Ξπ΅πΆπ΄
ΞπΆπ΄π΅ΞπΆπ΅π΄
X
ZY
Ξπππ
Your TurnGive all six names of the triangles
P R
Q
Ξπππ
Ξππ πΞπ ππΞπ ππΞπππ
Ξππ π
D
FE
Ξπ·πΈπΉΞπ·πΉπΈ ΞπΈπ·πΉ
ΞπΈπΉπ·ΞπΉπ·πΈΞπΉπΈπ·
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).
β π΄ corresponds to β π· since they are opposite the longest side of their triangles
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).
π΅πΆ corresponds to πΈπΉ since they are opposite the largest angle of their triangles
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
Vocabularyβ’ 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’ What is the includedSide for β π and β π ?
ππ is the included side of β π and β π
Vocabulary
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
β’ What is the included angleof ππ and π π?
β π is the included angle of ππ and π π
Congruence
Anglesβ’ two angles are congruent if they have the same measure (degrees)β’ β π΄ β β π΅ iff πβ π΄ = πβ π΅
Congruence
β’ Segments (sides)β’ two line segments are congruent if they have the same lengthβ’ π΄π΅ β πΆπ· iff π΄π΅ = πΆπ·
CongruenceTriangles
two triangles are congruentβ’ If each angle in one triangle is congruent to its
corresponding angle in the other triangle β’ AND β’ if each side in one triangle is congruent to its
corresponding side in the other triangle.
β’ In short we say βcorresponding parts of congruent triangles are congruentβ or βCPCTCβ
Congruence StatementsWhen stating congruence, the order is important
β’ 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 πβ’ β π΅ corresponds to β πβ’ β πΆ corresponds to β πβ’ π΄π΅ corresponds to ππβ’ π΅πΆ corresponds to ππβ’ πΆπ΄ corresponds to ππ
Your TurnEach pair of triangles is congruent:
Write a congruence statement for each pair of triangle.
Ξπ·πΈπΉ β _______ Ξπ ππ β ΞπΊπΉπ»Ξππ π
Why are Ξπ ππ and Ξπππ congruent? (That is, how do we prove it?)
β’ All corresponding parts are congruent (CPCTC)
β’ This is just the definition of congruence
Do we need all 6 pairs of angles and sides to be congruent to prove the triangles are congruent?...
Triangle Congruence
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
We can prove Triangle Congruence using congruence of only three pairs of corresponding parts.
Side-Side-Side (SSS) Congruency Axiom: if all three pairs of 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,
β’ Ξπ΄π΅πΆ β Ξπ·πΈπΉ by SSS
β’ Angle-Side-Angle (ASA) Congruency Axiom: 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,
β’ Ξπ΄π΅πΆ β Ξπ·πΈπΊ by ASA
Side-Angle-Side (SAS) Congruency Axiom: if two pairs of corresponding sides and the pair of included angles are congruent, then the triangles are congruent.
β’ ππ β ππ
β’ β πππ β β π ππΉ
β’ ππ β ππΉ
β’ Therefore,
β’ Ξπππ β ΞππΉπ by SAS
β’ Angle-Angle-Side (AAS) Congruency Axiom: If two pairs of corresponding angles are congruent and one 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,
β’ Ξπππ β ΞπΉπ·πΈ by AAS
Your Turn
β’ 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)
Your Turn
β’ 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.
3) 4)
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)
Angle-Angle-Angle (AAA) Condition
AAA controls shape only, not size
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
Congruence Conditions
β’ Conditions that work (axioms)β’ SSSβ’ ASAβ’ SASβ’ AAS
β’ Conditions that do not workβ’ AAAβ’ ASS
Symbols that show congruence (without giving measures)
π΄π΅ β πΉπΊ
π΄π΅ β πΉπΊ
π΄πΆ β πΈπΉ
Symbols that show congruence (without giving measures)
Write a congruence statement that identifies the additional information needed to prove these two triangles are congruent by ASA.
β π΄ β β FWrite a triangle congruence statement: Ξπ΄π΅πΆ β ΞπΉπΊπΈ
Are the triangles congruent? If so, write a congruence statement for the two triangles, and identify why are they congruent?
Ξπ΄π΅πΆ β ΞπΉπΊπΈ ππ¦ π΄π΄π
The Triangle Sum Theorem
If the polygon is a triangle then the sum of the interior angles = 180Β°
πβ π΄ + πβ B + πβ C = 180Β°
The Triangle Sum Theorem
If the polygon is a triangle then the sum of the interior angles = 180Β°
πβ π΄ + πβ B + πβ C = 180Β°
60Β°
70Β°
πβ C = ?
πβ C = 180Β° - 60Β° - 70Β°
πβ C = 50Β°
The Triangle Sum Theorem
If the polygon is a triangle then the sum of the interior angles = 180Β°
πβ π΄ + πβ B + πβ C = 180Β°
80Β°
75Β°
π₯ = ?
3π₯ β 5 = 180Β° - 80Β° - 75Β°
3x - 5 3π₯ = 180Β° - 80Β° - 70Β°
3π₯ = 180Β° - 150Β°
3π₯ = 30Β° π₯ = 10