6-2 triangle congruence
TRANSCRIPT
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
Bฮ๐ด๐ต๐ถฮ๐ด๐ถ๐ตฮ๐ต๐ด๐ถ
ฮ๐ต๐ถ๐ด
ฮ๐ถ๐ด๐ตฮ๐ถ๐ต๐ด
X
ZY
ฮ๐๐๐
Your TurnGive all six names of the triangles
P R
Q
ฮ๐๐๐
ฮ๐๐ ๐ฮ๐ ๐๐ฮ๐ ๐๐ฮ๐๐๐
ฮ๐๐ ๐
D
FE
ฮ๐ท๐ธ๐นฮ๐ท๐น๐ธ ฮ๐ธ๐ท๐น
ฮ๐ธ๐น๐ทฮ๐น๐ท๐ธฮ๐น๐ธ๐ท
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).
โ ๐ด corresponds to โ ๐ท 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).
๐ต๐ถ corresponds to ๐ธ๐น since they are opposite the largest angle of their triangles
Your Turn:
1) What angle does โ ๐ด correspond to?
2) What angle does โ ๐ correspond to?
3) What side does ๐๐ correspond to?
4) What side does ๐ด๐ถ correspond to?
โ ๐
โ ๐ถ
๐ถ๐ต
๐๐
Vocabularyโข 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.
โข ๐ ๐ is the included side of โ ๐ and โ ๐
โข What is the includedSide for โ ๐ and โ ๐ ?
๐๐ is the included side of โ ๐ and โ ๐
Vocabulary
โข Included angle: If two sides of a triangle are given, the included angle is the angle formed by those two sides.
โข โ ๐ is the included angle of ๐ ๐ and ๐๐
โข What is the included angleof ๐๐ and ๐ ๐?
โ ๐ is the included angle of ๐๐ and ๐ ๐
Congruence
Anglesโข two angles are congruent if they have the same measure (degrees)โข โ ๐ด โ โ ๐ต iff ๐โ ๐ด = ๐โ ๐ต
Congruence
โข Segments (sides)โข two line segments are congruent if they have the same lengthโข ๐ด๐ต โ ๐ถ๐ท iff ๐ด๐ต = ๐ถ๐ท
CongruenceTriangles
two triangles are congruentโข If each angle in one triangle is congruent to its
corresponding angle in the other triangle โข AND โข if each side in one triangle is congruent to its
corresponding side in the other triangle.
โข In short we say โcorresponding parts of congruent triangles are congruentโ or โCPCTCโ
Congruence StatementsWhen stating congruence, the order is important
โข 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
โข For Example, ฮ๐ด๐ต๐ถ โ ฮ๐๐๐ becauseโฆ โข โ ๐ด corresponds to โ ๐โข โ ๐ต corresponds to โ ๐โข โ ๐ถ corresponds to โ ๐โข ๐ด๐ต corresponds to ๐๐โข ๐ต๐ถ corresponds to ๐๐โข ๐ถ๐ด corresponds to ๐๐
Your TurnEach pair of triangles is congruent:
Write a congruence statement for each pair of triangle.
ฮ๐ท๐ธ๐น โ _______ ฮ๐ ๐๐ โ ฮ๐บ๐น๐ปฮ๐๐ ๐
Why are ฮ๐ ๐๐ and ฮ๐๐๐ congruent? (That is, how do we prove it?)
โข All corresponding parts are congruent (CPCTC)
โข 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) โ ๐ท is the included angle of which two sides?
2) What is the included angle of sides ๐ท๐น and ๐ธ๐น?
3) ๐ท๐น is the included side of which two angles?
4) What is the included side of โ ๐ท and โ ๐ธ
๐ท๐น and ๐ท๐ธ
โ ๐น
โ ๐ท and โ ๐น
๐ท๐ธ
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
โข ๐ด๐ต โ ๐ท๐ธ
โข ๐ต๐ถ โ ๐ธ๐น
โข ๐ถ๐ด โ ๐น๐ท
โข Therefore,
โข ฮ๐ด๐ต๐ถ โ ฮ๐ท๐ธ๐น by SSS
โข Angle-Side-Angle (ASA) Congruency Axiom: if two angles and their included side are congruent, then the two triangles are congruent.
โข โ ๐ด๐ต๐ถ โ โ ๐ท๐ธ๐บ
โข ๐ต๐ถ โ ๐ธ๐บ
โข โ ๐ต๐ถ๐ด โ โ ๐ธ๐บ๐ท
โข Therefore,
โข ฮ๐ด๐ต๐ถ โ ฮ๐ท๐ธ๐บ 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.
โข ๐๐ โ ๐๐
โข โ ๐๐๐ โ โ ๐ ๐๐น
โข ๐๐ โ ๐๐น
โข Therefore,
โข ฮ๐๐๐ โ ฮ๐๐น๐ by SAS
โข 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.
โข โ ๐๐๐ โ โ ๐ธ๐น๐ท
โข โ ๐๐๐ โ โ ๐น๐ท๐ธ
โข ๐๐ โ ๐น๐ธ
โข Therefore,
โข ฮ๐๐๐ โ ฮ๐น๐ท๐ธ by AAS
Your Turn
โข Determine which congruence condition proves the congruence for each the following pairs of triangles.
โข Write a congruence statement for each of the following pairs of triangles.
1) 2)
Your Turn
โข Determine which congruence condition proves the congruence for each the following pairs of triangles.
โข 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?
โข AAA
โข 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
Letโs look at an example of ASS
โข โ ๐ด โ โ ๐ท
โข ๐ด๐ต โ ๐ท๐ธ
โข ๐ต๐ถ โ ๐ธ๐น
Congruence Conditions
โข Conditions that work (axioms)โข SSSโข ASAโข SASโข AAS
โข Conditions that do not workโข AAAโข ASS
Symbols that show congruence (without giving measures)
๐ด๐ต โ ๐น๐บ
๐ด๐ต โ ๐น๐บ
๐ด๐ถ โ ๐ธ๐น
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.
โ ๐ด โ โ FWrite a triangle congruence statement: ฮ๐ด๐ต๐ถ โ ฮ๐น๐บ๐ธ
Are the triangles congruent? If so, write a congruence statement for the two triangles, and identify why are they congruent?
ฮ๐ด๐ต๐ถ โ ฮ๐น๐บ๐ธ ๐๐ฆ ๐ด๐ด๐
The Triangle Sum Theorem
If the polygon is a triangle then the sum of the interior angles = 180ยฐ
๐โ ๐ด + ๐โ B + ๐โ C = 180ยฐ
The Triangle Sum Theorem
If the polygon is a triangle then the sum of the interior angles = 180ยฐ
๐โ ๐ด + ๐โ B + ๐โ C = 180ยฐ
60ยฐ
70ยฐ
๐โ C = ?
๐โ C = 180ยฐ - 60ยฐ - 70ยฐ
๐โ C = 50ยฐ
The Triangle Sum Theorem
If the polygon is a triangle then the sum of the interior angles = 180ยฐ
๐โ ๐ด + ๐โ B + ๐โ C = 180ยฐ
80ยฐ
75ยฐ
๐ฅ = ?
3๐ฅ โ 5 = 180ยฐ - 80ยฐ - 75ยฐ
3x - 5 3๐ฅ = 180ยฐ - 80ยฐ - 70ยฐ
3๐ฅ = 180ยฐ - 150ยฐ
3๐ฅ = 30ยฐ ๐ฅ = 10