Chapter 4 Notes. 4.1 – Triangles and Angles A Triangle Three segments joining three noncollinear...
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Transcript of Chapter 4 Notes. 4.1 – Triangles and Angles A Triangle Three segments joining three noncollinear...
A Triangle Three segments joining three noncollinear points. Each point is a VERTEX of the triangle. Segments are SIDES!
A
B
C
Scalene – No congruent sides
Isosceles – At LEAST 2 congruent sides
Equilateral – All sides congruent
Acute – 3 acute angles
Obtuse – One obtuse angle
Right – One right angle
Equiangular – all angles congruent
8013m2m1m:Prove
ABC :Given
1 3
TRIANGLE SUM THEOREM
The sum of the measures of the angles of a triangle is 180.
2
Exterior Angles Theorem
The measure of an exterior angle of a triangle equals the sum of the two remote interior angles. (remote means nonadjacent)
1 2
3
4
Statement Reason
4m3m1m :Prove
1 angle
exterior with A triangle :Given
Corollary to triangle sum theorem: Acute angles of a right triangle are complementary.
All angles 180, if one is 90, the other two add up to 90, and are complementary
When TWO POLYGONS have the same size and shape, they are called CONGRUENT!
Their vertices and sides must all match up to be congruent.
When two figures are congruent, their corresponding sides and corresponding angles are congruent. Identical twins!
A
C
B
D
E
F
Name all the corresponding parts and sides, then make a congruence statement.
FEAB EDBC DFCA
FA EB DC
FEDABC BCACABEDFDFE
If you notice, the way you name the triangle is important, all the CORRESPONDING SIDES must line
up!
3rd Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the 3rd angles are congruent.
A
B
C D
E
F
FCThen
EBDAIf
,
• Note, triangles also have the following properties of congruent: Reflexive, symmetric, and transitive.
XYZABC
thenXYZDEF DEF,ABC :Transitive
ABC ,ABC :Symmetric
ABCABC :Reflexive
DEFDEF
A
C
B
D
E
FSSS Congruence Postulate – If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
FEDABCThen
DFACDEBCEFABIf
,,
FEDABCThen
DFACDCDEBCIf
,,
A
C
B
D
E
F
SAS Congruence Postulate – If two sides and the included
angle of one triangle are congruent to two sides and
the included angle of a second triangle, then the two
triangles are congruent.
Included means IN BETWEEN
A
C
B
D
E
FASA Congruence Postulate –
If two angles and the included side of one triangles are congruent to two angles and the included side of a
second triangle, then the two triangles are congruent.
FEDABCThen
DCDFACFAIf
,,
FEDABCThen
DFACDCEBIf
,,
A
C
B
D
E
F
AAS Congruence Theorem – If two angles and a nonincluded
side of one triangle are congruent to two angles and
the corresponding nonincluded side of a second triangle, then the two trianges
are congruent.
Helpful things for the future!Reflexive sides Reflexive angles
A
B
DC F
E
HG
BCBC EE
When you see shapes sharing a side, you state that fact using the reflexive property of
congruence!
Proofs! The way I like to think about it to look at all the angles and sides, and don’t be fooled by the picture.
A
B
C
D
E
EDCABC :Prove
BD and AE of
midpoint theis C:Given
BD and AE of
midpoint theis C 1. Given 1.
Tips, label the diagram as you go along.
Use SSS Congruence Postulate to show that DEFABC
A
C
B
DE
F
(-3, -2)
(-4, -3)
(-5, 1)
(2, 2)
(5, 4)(1, 3)
AC
BC
AB
DF
EF
DE
A
B
D
E
EDBABD :Prove
DE ||AB,BE ||AD :Given
DE ||AB
BE ||AD 1. Given 1.
Tips, label the diagram as you go along.
A
B
C
D
E
ED||AB :Prove
BD and AE of
midpoint theis C:Given
BD and AE of
midpoint theis C 1. Given 1.
DCBC
ECAC 2.
mdpt of Def 2.
ECDACB 3. VAT 3.ECDACB 4. post SAS 4.
are s' of Parts CorrespCongruent are
TrianglesCongruent ofPart ingCorrespondCPCTC
Some Ideas that may help you.
If they want you to prove something, and you see triangles in the picture, proving triangles to be congruent may be helpful.
If they want parallel lines, look to use parallel line theorems (CAP, AIAT, AEAT, CIAT)
Know definitions (Definition of midpoint, definition of angle bisectors, etc.)
Sometimes you prove one pair of triangles are congruent, and then use that info to prove another pair of triangles are congruent.
• Bring book Tuesday
• We will go over what’s going to be on Wednesday’s Quiz at end of Tuesday lesson
Vertex Angle
Base Angles
BASE
LEGS
Remember, definition of isosceles triangles is that AT LEAST two congruent sides.
Base angles theorem – If two sides of a triangle are congruent, then the base angles are congruent.
Converse of Base angles theorem – If base angles are congruent, then the two opposite sides are congruent.
Corollary 1 – An equilateral triangle is also equiangular (Use isosceles triangle theorem multiple times with transitive)
Corollary 2 – An equilateral triangle has three 60 degree angles (Use corollary 1 and angle of triangle equals 180)
Hypotenuse Leg Theorem (HL) – If the hypotenuse and ONE of the legs of a RIGHT triangle are congruent, then the triangles are congruent.
D
U C K1 2
KUove
anglesrightareand
legsasDKandDU
withtriangleisoscelesanisDUKGiven
:Pr
.21
.
:
DKDU Def of isosceles triangle
Given a right triangle with one vertex (-20, -10), and legs of 30 and 40, find two other vertices, then find the length of the hypotenuse.
Given a vertex of a rectangle at the origin, find three other possible vertices if the base is 15 and the height is 10 for a rectangle. Then find the area.