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Triangles Part 1
The sum of the angles in a triangle is always equal to:
180°
Classification By Angle
AcuteA triangle that has all 3 acute angles
ObtuseA triangle with one obtuse angle and 2 acute angles
RightA triangle with 1 right angle and 2 acute angles
The two acute angles must = 90° therefore they are complimentary
EquiangularA triangle with all 3 angles congruent
They must each = 60 °
Classification by Sides
ScaleneAll three sides have different lengths
IsoscelesTwo sides have the same length
Equilateral All three sides have the same length
Isosceles Triangles
B
A C
AB = CB and <A = < C
Leg Leg
Base
Equilateral Triangle
An equilateral triangle is also equiangular.
An equiangular triangle is also equilateral
B
A C
AB = BC = AC <A = <B = <C
Classify each triangle by its angles and sides.
Equilateral
Scalene, Right
Isosceles, Acute
Isosceles, Obtuse
Scalene, Acute
Isosceles, Right
Using the Distance Formula to classify triangles by their sides
Find the measure of the sides of triangle DCE,
then classify the triangle by sides. D(3,9);E(−5,3);C(2,2)
Step 1: Find the distance of all three sides using the distacneformula. D = (x
2- x1)2 + (y
2- y1)2
DE = (-5 - 3)2 + (3-9)2= (-8)2+(-6)2 = 64+36 = 100 = 10
EC =
DC =
Step 2: Classify the traingle
Since _____ sides are congruent the traingle is called ______
You Try
Find the measure of the sides of RST. Classify the triangle by sides.
RST is Scalene
R(−1.−3);S(4,4);T(8,−1)
RS = 74;ST = 41;RT = 85
Find the missing Values
Find x and the measure of each side of an equilateral triangle RST if:
RS =x+ 9;ST =2x;RT =3x−9
2x
3x-9
x+9
Step 1: Draw and equilateraltraingle and label the giveninformation.
S
R T
3x-9 = 2x -9 = -x 9 = x
x+9 = 3x - 9 9 = 2x - 9 18 = 2x 9 = x
x+9 = 2x 9 = x
2x
3x-9
x+9
Step 1: Draw and equilateraltraingle and label the giveninformation.
Step 2: Set any 2 sidesequal to eachother andsolve for x. (It does notmatter which two sides youchoose since all three areequal)
S
R T
RT = 3x-9 RT = 3(9)-9RT = 27-9RT = 18
ST = 2x ST = 2(9)ST = 18
RS = x + 9 RS = 9 + 9 RS = 18
x = 9
2x
3x-9
x+9
Step 1: Draw and equilateral traingleand label the given information.
Step 2: Set any 2 sides equal toeachother and solve for x. (It doesnot matter which two sides youchoose since all three are equal)
Step 3: Plug x into one side to getall three side lengths. (To checkyour answer plug x into the othertwo sides and make sure all threesides are equal.)
S
R T
You Try
Find d and the measure of each side of an equilateral triangle KLM if:
KL =d+ 2;LM =12 −d;KM =4d−13
d =5;KL =LM =KM =7
One more! (This one is a little different)
Find x and the measure of all sides if COW is
isosceles, with CO=CW, and CO =x+ 7;CW =3x−5;OW =x−1
x =6;CO=13,CW =13,OW =5
Finding the Measure of Missing Angles
The sum of the angles in a triangle is always equal to:
180°
Examples
Find X:
39°65°
x40°
x
30°
2x
x x
1.)2.)
3.) 4.)
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles.
Exterior Angle: An angle formed when one side of a triangle is extended
Remote Interior Angles: The interior angles of the triangle that are not adjacent to the exterior angle
m∠1 + m∠2 = m∠4
∠4 is an exteriorangle
∠1 & ∠2 are remote interior angles to∠4
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2
1
Proof of Exterior Angle Theorem
1.) m∠1 + m∠2 + m∠3 = 180 By Def of a Triangle
2.) m∠3 + m∠4 = 180 By Def of Liner Pair
3.) m∠1 + m∠2 + m∠3 = m∠3 + m∠4 By Substitution
4.) m∠1 + m∠2 = m∠4 By SPOE
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Bigger Picture
Find all missing angles
5
4 32
1
38°
32°
41°64°