triangle fundamentals intro to g.10 modified by lisa palen 1
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TriangleFundamentals
Intro to G.10
Modified by Lisa Palen 1
Triangle
A
B
C
Definition: A triangle is a three-sided polygon.
What’s a polygon?
2
These figures are not polygons These figures are polygons
Definition: A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints.
Polygons3
Definition of a Polygon
A polygon is a closed figure in a plane formed by a finite number of segments that intersect only at their endpoints.
4
Triangles can be classified by:
Their sidesScaleneIsoscelesEquilateral
Their anglesAcuteRightObtuse Equiangular
5
Classifying Triangles by Sides
Equilateral:
Scalene: A triangle in which no sides are congruent.
Isosceles:
AB
= 3
.02
cm
AC
= 3.15 cm
BC = 3.55 cm
A
B CAB =
3.47
cmAC = 3.47 cm
BC = 5.16 cmBC
A
HI = 3.70 cm
G
H I
GH = 3.70 cm
GI = 3.70 cm
A triangle in which at least 2 sides are congruent.
A triangle in which all 3 sides are congruent.
6
Classifying Triangles by Angles
Obtuse:
Right:
A triangle in which one angle is....
A triangle in which one angle is...
108
44
28 B
C
A
34
56
90B C
A
obtuse.
right.
7
Classifying Triangles by Angles
Acute:
Equiangular:
A triangle in which all three angles are....
A triangle in which all three angles are...
acute.
congruent.
57 47
76
G
H I
8
Classificationof Triangles
with
Flow Chartsand
Venn Diagrams9
polygons
Classification by Sides
triangles
Scalene
Equilateral
Isosceles
Triangle
Polygon
scalene
isosceles
equilateral
10
polygons
Classification by Angles
triangles
Right
Equiangular
Acute
Triangle
Polygon
right
acute
equiangular
Obtuse
obtuse
11
Naming Triangles
For example, we can call this triangle: A
B
C
We name a triangle using its vertices.
∆ABC
∆BAC
∆CAB ∆CBA
∆BCA
∆ACBReview: What is ABC?
12
Parts of Triangles
For example, ∆ABC has
Sides: Angles: A
B
C
Every triangle has three sides and three angles.
ACB
ABC
CABABBCAC
13
Opposite Sides and Angles
A
B C
Opposite Sides:
Side opposite of BAC :
Side opposite of ABC :
Side opposite of ACB :
Opposite Angles:
Angle opposite of : BAC
Angle opposite of : ABC
Angle opposite of : ACB
BC
AC
AB
BC
AC
AB
14
Interior Angle of a Triangle
For example, ∆ABC has interior angles:
ABC, BAC, BCA
A
B
C
An interior angle of a triangle (or any polygon) is an angle inside the triangle (or polygon), formed by two adjacent sides.
15
Interior Angles
Exterior Angle
For example, ∆ABC has exterior angle ACD, because ACD forms a linear pair with ACB.
An exterior angle of a triangle (or any polygon) is an angle that forms a linear pair with an interior angle. They are the angles outside the polygon formed by extending a side of the triangle (or polygon) into a ray.
A
BC
D
Exterior Angle
16
Interior and Exterior Angles
For example, ∆ABC has exterior angle:
ACD and
remote interior angles A and B
The remote interior angles of a triangle (or any polygon) are the two interior angles that are “far away from” a given exterior angle. They are the angles that do not form a linear pair with a given exterior angle.
A
BC
D
Exterior AngleRemote Interior Angles
17
Triangle Theorems
18
Triangle Sum Theorem
The sum of the measures of the interior angles in a triangle is 180˚.
m<A + m<B + m<C = 180IGO GeoGebra Applet
19
Third Angle Corollary
If two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent.
20
Third Angle Corollary Proof
The diagramGiven:
statements reasons
E
DA
B
CF
Prove: C F
1. A D, B E2. mA = mD, mB = mE3. mA + mB + m C = 180º mD + mE + m F = 180º4. m C = 180º – m A – mB m F = 180º – m D – mE5. m C = 180º – m D – mE6. mC = mF 7. C F
1. Given2. Definition: congruence3. Triangle Sum Theorem
4. Subtraction Property of Equality
5. Property: Substitution6. Property: Substitution7. Definition: congruenceQED
21
Corollary
Each angle in an equiangular triangle measures 60˚.
60
6060
22
CorollaryThere can be at most one right or obtuse angle in a triangle.
Example
Triangles???
23
CorollaryAcute angles in a right triangle are complementary.
Example
24
Exterior Angle Theorem
The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
Exterior AngleRemote Interior Angles A
BC
D
m ACD m A m B
Example:
(3x-22)x80
B
A DC
Find the mA.
3x - 22 = x + 80
3x – x = 80 + 22
2x = 102
x = 51
mA = x = 51°
25
Exterior Angle TheoremThe measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
GeoGebra Applet (Theorem 1)
26
Special Segmentsof Triangles
27
Introduction
There are four segments associated with triangles:
Medians Altitudes Perpendicular Bisectors Angle Bisectors
28
Median - Special Segment of Triangle
Definition: A segment from the vertex of the triangle to the midpoint of the opposite side.
Since there are three vertices, there are three medians.
In the figure C, E and F are the midpoints of the sides of the triangle.
, , .DC AF BE are the medians of the triangle
B
A DE
C F
29
Altitude - Special Segment of Triangle
Definition: The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side.
In a right triangle, two of the altitudes are the legs of the triangle.
B
A DE
C
FB
A D
F
In an obtuse triangle, two of the altitudes are outside of the triangle.
, , .AF BE DC are the altitudes of the triangle
, ,AB AD AF altitudes of right B
A D
F
I
K , ,BI DK AF altitudes of obtuse
30
Perpendicular Bisector – Special Segment of a triangle
AB PR
Definition: A line (or ray or segment) that is perpendicular to a segment at its midpoint.
The perpendicular bisector does not have to start from a vertex!
Example:
C D
In the scalene ∆CDE, is the perpendicular bisector.
In the right ∆MLN, is the perpendicular bisector.
In the isosceles ∆POQ, is the perpendicular bisector.
EA
B
M
L N
A BR
O Q
P
AB
31
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