2. 3.6 T YPES OF T RIANGLES Made By: Brad Smertz, Blair Cacciamani, & Ricky Guditus

3. T YPES OF T RIANGLES – I SOSCELES 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".right angle The altitude is a perpendicular distance from the base to the topmost vertex. *Refer to the following link --- http://www.mathopenref.com/isosceles.html http://www.mathopenref.com/isosceles.html

4. T YPES OF T RIANGLES - S CALENE 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

5. T YPES OF T RIANGLES - E QUILATERAL Definition – An equilateral triangle is a triangle in which all sides are congruent

6. T YPES OF T RIANGLE - E QUIANGULAR Definition: An equiangular triangle is a triangle in which all angles are congruent

7. T YPE OF T RIANGLES - A CUTE 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

8. T YPES OF T RIANGLES - R IGHT 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°

9. T YPES OF T RIANGLES - O BTUSE 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°

10. S AMPLE P ROBLEMS ! Q: What type of triangle is shown above? A: The triangle is an equilateral triangle because all sides are equal.

11. S AMPLE P ROBLEMS ! ( 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.

12. S AMPLE P ROBLEMS ( 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.

13. S AMPLE P ROBLEMS (C ONT ’ 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.)

14. P RACTICE P ROBLEMS ! Problems --- Classify each triangle shown or described as equilateral, isosceles, or scalene. 1.) 2.) 3.) 4.) Equilateral Isosceles Scalene

15. P RACTICE P ROBLEMS ! (C ONT ’ D ) 1.) Δ ABC 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.) Δ DEF with DE = 6, EF = 8, and DF = 10. Therefore, the triangle is clearly scalene because no sides are equal.

16. P RACTICE P ROBLEMS (C ONT ’ D ) Directions: Identify the triangle type below --- 1.) 2.) 3.) 4.) Equiangular Right Obtuse Acute

17. A LWAYS, S OMETIMES, N EVER !? 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!!!

18. S AMPLE P ROOFS

19. P ROOF Given: 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)

20. S AMPLE : A LGEBRAIC P ROBLEM 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 2 x = 84° Þ x = 42° Answer : The sizes of the other two angles are 42º each.

21. A LGEBRAIC P ROBLEM : 1 A right triangle has one other angle that is 45º. Besides being right triangle what type of triangle is this? C AB

22. A LGEBRAIC P ROBLEM : 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.

23. A LGEBRAIC P ROBLEM :2 Given: P = 50. Segment AB segment CB Segment AC is 5 more then segment AB Find: a. Length of segment AC b. Is ABC isosceles?

24. A LGEBRAIC P ROBLEM : 2 50 = x + x + x + 5 50 = 3 x + 5 3 x = 50 – 5 3 x = 45 x =15 15 + 5 =20 Side 1- x Side 2- x Side 3- x+5 Answer: a. Segment AC = 20 b. Yes

25. 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.