3.6 TYPES OF TRIANGLES

Made By: Brad Smertz, Blair Cacciamani, & Ricky Guditus

TYPES OF TRIANGLES – ISOSCELES

Definition: An isosceles triangle is a triangle in which at least two sides are congruent

Properties of Isosceles Triangles -• The unequal side of an isosceles triangle is usually referred to as the 'base' of the triangle.• The base angles of an isosceles triangle are always equal. In the figure above, the

angles ∠BAC and ∠ACB are always the same• When the 3rd angle is a right angle, it is called a "right isosceles triangle".• The altitude is a perpendicular distance from the base to the topmost vertex.

*Refer to the following link --- http://www.mathopenref.com/isosceles.html

TYPES OF TRIANGLES - SCALENE Definition – A scalene triangle is a triangle in

which no sides are congruent. Properties of Scalene Triangles –• Interior angles are all different• Shortest side is opposite the smallest angle• Longest side is opposite the largest angle

TYPES OF TRIANGLES - EQUILATERAL

Definition – An equilateral triangle is a triangle in which all sides are congruent

TYPES OF TRIANGLE- EQUIANGULAR

Definition: An equiangular triangle is a triangle in which all angles are congruent

TYPE OF TRIANGLES - ACUTE

Definition – An acute triangle is a triangle in which all angles are acute

Acute angle: An angle whose measure is less than 90°

http://www.mathopenref.com/angleacute.html

TYPES OF TRIANGLES - RIGHT

Definition – A right triangle is a triangle in which one of the angles is a right angle. (The side opposite the right angle is called the hypotenuse. The sides that form the right angle are called legs.)

Right angle: An angle whose measure is exactly 90°

TYPES OF TRIANGLES - OBTUSE

Definition – An obtuse triangle is a triangle in which one of the angles in an obtuse angle

Obtuse Angle: An angle whose measure is greater than 90° and less than 180°

SAMPLE PROBLEMS!

Q: What type of triangle is shown above?A: The triangle is an equilateral triangle because all sides are equal.

SAMPLE PROBLEMS! (CONT’D)

Q: What type of triangle is shown above?A: The triangle shown above is an isosceles triangle because at least two sides are congruent.

SAMPLE PROBLEMS (CONT’D)

Q: What type of triangle is portrayed above?

A: The triangle pictures above is an acute triangle because all three angles (<ACB, <ABC, and <BAC) of the triangle are acute (less than 90 degrees.)

Q: Name the type of triangle.

A: The triangle shown above is an obtuse triangle because at least one of the three angles in the triangle is obtuse (greater than 90 degrees.)

Q: Identify the triangle above.

A: The triangle above is classified as a scalene triangle because no sides are congruent.

SAMPLE PROBLEMS (CONT’D)

Q: What type is triangle is pictured above?

A: The triangle above is dubbed as a right triangle because at least one angle is a right angle (90 degrees.)

Q: Identify the triangle above.

A: The triangle above is classified as an equiangular triangle because all three of the angles are congruent (All angles have the same measure of 60 degrees.)

PRACTICE PROBLEMS!Problems ---

Classify each triangle shown or described as equilateral, isosceles, or scalene.

1.) 2.)

3.)

4.)

Equilateral Isosceles

Scalene Scalene

PRACTICE PROBLEMS! (CONT’D)

1.) ΔABC with AB = 10, BC = 10, and AC = 8.

*Draw a diagram if necessary* Therefore, the triangle is isosceles because at least two sides are equal.

2.) ΔDEF with DE = 6, EF = 8, and DF = 10.

Therefore, the triangle is clearly scalene because no sides are equal.

PRACTICE PROBLEMS (CONT’D)

Directions: Identify the triangle type below ---

1.) 2.)

3.)4.)

Equiangular

Right

Obtuse

Acute

ALWAYS, SOMETIMES, NEVER!?

1.) An isosceles triangle may also be a right triangle.

Sometimes!!!

2.) It is possible for an obtuse triangle to be isosceles.

Sometimes!!!

3.) A triangle cannot have more than one right angle.

Always!!!

SAMPLE PROOFS

PROOFGiven:

1.) Segment QR is congruent to segment ST

2.) Segment UR is congruent to segment US

3.) Segment QS is congruent to segment RT

4.) Angle 3 is congruent to angle 2

5.) Triangle QUS is congruent to triangle TUR

Prove:

1.) Given

2.) Given

3.) Addition

4.) If sides, then angles

5.) SAS (2,3,4)

SAMPLE :ALGEBRAIC PROBLEM

Example: An isosceles triangle has one angle of 96º. What are the sizes of the other two angles?

Solution:Step 1: Since it is an isosceles triangle it will have two equal angles. The given 96º angle cannot be one of the equal pair because a triangle cannot have two obtuse angles..

Step 2: Let x be one of the two equal angles. The sum of all the angles in any triangle is 180°.       x + x + 96° = 180° Þ 2x = 84° Þ x = 42°

Answer: The sizes of the other two angles are 42º each.

ALGEBRAIC PROBLEM: 1

A right triangle has one other angle that is 45º. Besides being right triangle what type of triangle is this?

C

A B

Isosceles Triangle!

ALGEBRAIC PROBLEM: 1 ANSWER

Solution:Step 1: Since it is right triangle it will have

one 90º angle. The other angle is given as 45º.

Step 2: Let x be third angle. The sum of all the angles in any triangle is 180º.      x + 90º + 45º = 180° x = 45º

Step 3: Two of the angles are equal which means that it is an isosceles triangle.

Answer: It is also an isosceles triangle.

ALGEBRAIC PROBLEM:2

Given:P = 50.Segment AB segment CBSegment AC is 5 more then segment AB

Find:a. Length of segment AC b. Is ABC isosceles?

ALGEBRAIC PROBLEM: 2

50 = x + x + x+ 5 50 = 3x + 5 3x = 50 – 5

3x = 45x =15

15 + 5 =20

Side 1- xSide 2- xSide 3- x+5

Answer: a. Segment AC = 20 b. Yes

Works Cited

“Acute Angle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

“Acute Triangle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

“Equiangular Triangle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

“Equilateral Triangle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

“Isosceles Triangle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

“Isosceles, Equilateral, Scalene, Obtuse.” Types of Triangles. Math Warehouse, 2002. Web. 15 January 2011.

“Obtuse Angle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

“Obtuse Triangle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

“Problems.” Types of Triangles Practice Problems. Education.com, 2006. Web. 19 January 2011.

“Right Angle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

“Right Triangle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

“Scalene Triangle.” Definition. Math Open Reference, 2009. Web. 15 January 2011.

Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. Boston: McDougal, Littell & Company, 1991. Print.