unit 3
DESCRIPTION
Unit 3. Triangles and their properties. Lesson 3.1. Classifying Triangles Triangle Sum Theorem Exterior Angle Theorem. Classification of Triangles by Sides. Classification of Triangles by Angles. Example 1. You must classify the triangle as specific as you possibly can. - PowerPoint PPT PresentationTRANSCRIPT
Lesson 3.1
Classifying TrianglesClassifying Triangles
Triangle Sum TheoremTriangle Sum Theorem
Exterior Angle TheoremExterior Angle Theorem
Classification of Triangles by Sides
NameName EquilateralEquilateral IsoscelesIsosceles ScaleneScalene
Looks LikeLooks Like
CharacteristicsCharacteristics 33 congruent congruent sidessides
At At least 2least 2 congruent sidescongruent sides
NoNo Congruent Congruent SidesSides
Classification of Triangles by Angles
NameName AcuteAcute EquiangularEquiangular RightRight ObtuseObtuse
Looks LikeLooks Like
CharacteristicsCharacteristics 33 acuteacute anglesangles
33 congruentcongruent anglesangles
11 rightright anglesangles
11 obtuseobtuse angleangle
Example 1
You must classify the triangle as specific as You must classify the triangle as specific as you possibly can.you possibly can.
That means you must nameThat means you must name Classification according to anglesClassification according to angles Classification according to sidesClassification according to sides
In that order!In that order! ExampleExample
Obtuse isosceles
More Examples
8.6 8.6
8.6
600
600
600
2.5 2.5
4.5
260 260
1280
Equiangular
Equilateral
Obtuse
Isosceles
And more examples…Draw a sketch of the following triangles. Use proper symbol notation
Obtuse Scalene Equilateral Right Equilateral
impossible
Proving the Sum of a triangle’s Angles
What do we know about the two green angles labeled X?
What do we know about the two yellow angles labeled Y?
What do we know about the three angles at the top of the triangle? (the X, Y and Z)
Proving the Sum of a triangle’s Angles
X=X because of Alternate interior angles.
What do we know about the two yellow angles labeled Y?
What do we know about the three angles at the top of the triangle? (the X, Y and Z)
Proving the Sum of a triangle’s Angles
X=X because of Alternate interior angles.
Y=Y because of Alternate interior angles.
What do we know about the three angles at the top of the triangle? (the X, Y and Z)
Proving the Sum of a triangle’s Angles
X=X because of Alternate interior angles.
Y=Y because of Alternate interior angles.
X+Z+Y=180 because they form a straight line.
Proving the Sum of a triangle’s Angles
X=X because of Alternate interior angles.
Y=Y because of Alternate interior angles.
X+Z+Y=180 because they form a straight line.
The sum of the interior angles must also be equal to 1800
Triangle Sum Theorem
The sum of the interior angles of a triangle The sum of the interior angles of a triangle is 180.is 180.
Proving the Exterior Angle Theorem
180m a m b c triangle sumtheorem
180m b m d Linear Pair a+b+c=b+d Substitution
a+c=d Subtraction (subtract b from both sides)
Side/Angle Pairs in a Triangle
A
B
C
Angle A and the side opposite it are a pair
Angle B and the side opposite it are a pair
Angle C and the side opposite it are a pair
The Inequalities in One Triangle
If it is the longest side, then it is opposite If it is the longest side, then it is opposite largest angle measure.largest angle measure.
If it is the shortest side, then it is opposite If it is the shortest side, then it is opposite the smallest angle measure.the smallest angle measure.
If it is the middle length side, then it is If it is the middle length side, then it is opposite the middle angle measure.opposite the middle angle measure.
and their converse, too!
If it is the largest angle measure, then it is If it is the largest angle measure, then it is opposite the longest side.opposite the longest side.
If it is the smallest angle measure, then it is If it is the smallest angle measure, then it is opposite the shortest side.opposite the shortest side.
If it is the middle angle measure, then it is If it is the middle angle measure, then it is opposite the middle length side.opposite the middle length side.
More examples…Order the angles from smallest to largest
12
11
5.8
In UVW
VW
UW
UV
U
V
W
5.8 12
11
W
V
U
Week’s Schedule
Mon: Lesson 3.3Mon: Lesson 3.3 Tue: Lesson 3.4Tue: Lesson 3.4 Wed: MEAPWed: MEAP Thu: Quiz/Lesson 3.5Thu: Quiz/Lesson 3.5 Fri: Practice testFri: Practice test
Mon: Review Unit 3Mon: Review Unit 3 Tue: Unit 3 TestTue: Unit 3 Test
Examples: solve for x and/or y
7
75
30
If the two angles are equal and the interior angles of a triangle have a sum of 180, what is the measure of the two angles?
3x = 45
x = 15
y+7 = 45
y = 38
One more example…solve for x and y
3x-11 = 2x+11x –11 = 11
x = 22
Using the value of x, the measure of the two angles are each… 55 degrees
Using the triangle sum theorem the last angle measure is… 70 degrees
2y = 70
y = 35
Prove what the angles in an equilateral triangle MUST always be.
If all the sides are the same, equilateral, then all the angles must be the same, equiangular.
If one of the angles is x, then all of the angles must also be x.
x + x + x = 180 (triangle sum)3x = 180 (combine like terms)x = 60 (DPOE)
One more!
All sides are equal. Pick any two and set them equal to each other. Then solve for x.
12x – 13 = 2x + 17
10x –13 = 17
10x = 30
x = 3
Lesson 3.4
Altitudes, Medians, and Perpendicular Altitudes, Medians, and Perpendicular Bisectors of TrianglesBisectors of Triangles
Putting old terms together…
Perpendicular:Perpendicular:
Two lines that intersect at a right angle.Two lines that intersect at a right angle. Bisector:Bisector:
A segment, ray, or line that divides a segment into A segment, ray, or line that divides a segment into two congruent parts.two congruent parts.
Perpendicular bisector (of a triangle):Perpendicular bisector (of a triangle):
A segment, ray, or line that is perpendicular to a A segment, ray, or line that is perpendicular to a side of a triangle at the midpoint of the side.side of a triangle at the midpoint of the side.
Is segment BD a perpendicular bisector? Explain!
No, it is nether perp. nor a bisector.
Yes, it is perp. to segment AC and divides it into two congruent parts.
No, it is a bisector but is not perp.
No, segment BD is perp. to segment BC, but is not its bisector
New vocabulary terms!Median of a TriangleMedian of a Triangle: A segment whose : A segment whose endpoints are a vertex of the triangle and the endpoints are a vertex of the triangle and the midpoint of the opposite side.midpoint of the opposite side.
Altitude of a TriangleAltitude of a Triangle: The perpendicular : The perpendicular segment from a vertex to the opposite side or segment from a vertex to the opposite side or to the line that contains the opposite sideto the line that contains the opposite side
Is segment BD a Median? Altitude? Explain!
Neither, it is not a bisector and it is not perp.
Both, it is a bisector and is perp.
Median, it is a bisector of segment AC
DAltitude, it is perp. to segment BC
Special notes about Perp. Bisectors and Medians: All perp. bisectors are also medians.All perp. bisectors are also medians. Some medians are perp. bisectors.Some medians are perp. bisectors. If it’s not a median, then it is not a perp. If it’s not a median, then it is not a perp.
bisector.bisector.
Special notes about Perp. Bisectors and Altitudes: All perp. bisectors are also altitudes.All perp. bisectors are also altitudes. Some altitudes are perp. bisectors.Some altitudes are perp. bisectors. If it’s not a altitude, then it is not a perp. If it’s not a altitude, then it is not a perp.
bisector.bisector.
Area Formula of a Triangle
1
2A b h
b is the base
h is the height
HINT: The base and the height always meet at a right angle
Formula for the perimeter of a triangle.P=a+b+cP=a+b+c
a, b, and c are the three sides of the triangle.a, b, and c are the three sides of the triangle.
HINT: Perimeter is the sum of all three sides HINT: Perimeter is the sum of all three sides of a trianlgeof a trianlge