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

43

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

43

2

1

Bigger Picture

Find all missing angles

5

4 32

1

38°

32°

41°64°