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Unit 4 Notes: Triangles
4-1 Triangle Angle-Sum Theorem
Angle review, label each angle with the correct classification:
___________________ _____________________ _______________________
Triangle – a polygon with three sides.
There are two ways to classify triangles: by angles and by sides
There are four ways to classify a triangle by its angles:
________________ __________________ _________________ __________________
There are three ways to classify triangles based on sides
____________ _________________ _______________
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Can triangles be named in the following ways? Yes/No, Why?
Acute Scalene ___________________________________
Isosceles Right __________________________________
Acute Equilateral ____________________________
Obtuse Equilateral ________________________________
Right Equilateral _________________________________
Statement Reason
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
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Triangle Exterior Angle Theorem
side side
Remote
remote
exterior
2
m<3
4
Practice, find the measure of each variable
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4-2 Triangle Congruence & Third Angle Theorem
Congruent Polygons – If two polygons are congruent, then their corresponding parts are congruent.
The converse can be used to prove two figures are congruent.
Examples:
Third-Angle-Theorem
If two angles of one triangle are congruent to two angles in another triangle, then the third angles in the
triangles are also congruent.
Examples:
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B
H
M A
T
Given:
B is the midpoint of Prove:
Statement Reason
1. _______________________________________ 1. _____________________________________
2. _______________________________________ 2. _____________________________________
3. _______________________________________ 3. _____________________________________
4. _______________________________________ 4. _____________________________________
5. _______________________________________ 5. _____________________________________
6. _______________________________________ 6. _____________________________________
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4-3 SSS & SAS Congruence Postulates
Side – Side – Side Triangle Congruence Postulate (SSS)
If all three sides of one triangle are congruent to three sides of another triangle, then the triangles are
congruent.
Examples:
Given: , B is the midpoint of AC
Prove: CBDABD
BA C
D
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Side – Angle – Side Triangle Congruence Postulate (SAS)
If two sides and an included angle of one triangle are congruent to two sides and an included angle
of another triangle, then the two triangles are congruent.
Included Angle - The angle between two sides of a triangle.
Examples:
Given: C is the midpoint of AE and BD
Prove:
D
C
A
E
B
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4-4 ASA & AAS Conguence
Angle – Side – Angle Triangle Congruence Postulate (ASA)
If two consecutive angles and the included side of a triangle are congruent to two consecutive angles and the
included side of another triangle, then the triangles are congruent.
Examples:
Given: C is the midpoint of AE , A E
Prove:
D
C
A
E
B
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Angle – Angle – Side Triangle Congruence Theorem (AAS)
If two consecutive angles and the corresponding side of one triangle are congruent to two consecutive
angles and the corresponding side of another triangle, then the triangles are congruent.
*This theorem stems from the third angle theorem.
Examples:
Given: ACDB ,
Prove: CBDABD
BA C
D
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4-5 Corresponding parts of congruent triangles are congruent CPCTC
Corresponding parts of congruent figures / triangles are congruent – CPCFC or CPCTC
If two figures are congruent, then all corresponding parts of those Figures/Triangles are congruent.
Examples:
Given: The figure with AB AD , DC AD , AB = CD, and E is the midpoint of AD .
Prove:
C
EA D
B
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4-6 Base Angle Theorem
Base angle theorem for Isosceles triangles –
If two sides of a triangle are congruent, then the base angles are congruent.
Converse to the base angle theorem for isosceles triangles –
If the base angles of a triangle are congruent, then the sides opposite the base angles are congruent.
Isosceles vertex angle bisector theorem –
The bisector of a vertex angle of an isosceles triangle is perpendicular to the base of the triangle.
Corollaries- If a triangle is equilateral, then it is equiangular.
If a triangle is equiangular, then it is equilateral.
Examples:
Given: EBC ECB , AB = CD, and E is the midpoint of AD
Prove:
C
EA D
B
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4-7 Hypotenuse Leg (HL) Theorem & Leg Leg (LL) Theorem
Hypotenuse Leg (HL) Congruence Theorem -
If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right
triangle, then the triangles are congruent.
Leg Leg (LL) Congruence Theorem -
If the leg and leg of one right triangle are congruent to the leg and leg of another right triangle, then the
triangles are congruent. These is not used often because of the SAS Postulate
Examples:
Given: BCAB , BCDC , and DBAC
Prove: BAC CDB
E
D
B C
A