TRIANGLES

1. TRIANGLES PARTS, CLASSIFICATIONS, ANGLES NAD PROVING CONGRUENCE OF TRIANGLES .

2. PARTS OF TRIANGLES • Sides • the edges or boundaries of the triangle. • Vertices • part where the two sides join. • Adjacent sides • two sides that have common vertex

3. PARTS OF TRIANGLES • In a right triangle • Legs • the sides adjacent to the right angle in a right triangle. • Hypotenuse • the side opposite the right angle in a right angle.

4. PARTS OF TRIANGLES • In an isosceles triangle, • Legs -the congruent sides • Base -the side that is not congruent to any side of an isosceles triangle.

5. Different Types of Triangles • There are several different types of triangles. • You can classify a triangle by its sides and its angles. • There are THREE different classifications for triangles based on their sides. • There are FOUR different classifications for triangles based on their angles.

6. Classifying Triangles by Their Sides • EQUILATERAL – 3 congruent sides • ISOSCELES – at least two sides congruent • SCALENE – no sides congruent EQUILATERAL ISOSCELES SCALENE

7. Classifying Triangles by Their Angles • EQUIANGULAR – all angles are congruent • ACUTE – all angles are acute • RIGHT – one right angle • OBTUSE – one obtuse angle EQUIANGULAR ACUTE RIGHT OBTUSE

8. Congruent Triangles ROTATION REFLECTION TRANSLATION What is "Congruent" ... ? It means that one shape can become another using Turns, Flips and/or Slides:

9. Congruent Triangles If two triangles are congruent they will have exactly the same three sidesand exactly the same three angles. The equal sides and angles may not be in the same position (if there is a turn or a flip), but they will be there.

10. Same Sides of a Triangle because they all have exactly the same sides. • If the sides are the same then the triangles are congruent. • For example: is congruent to and

11. Same Sides of a Triangle because the two triangles do not have exactly the same sides. • If the sides are the same then the triangles are congruent. • For example: is not congruent to

12. Same Angles of a Triangle because, even though all angles match, one is larger than the other. Does this also work with angles? Not always! Two triangles can have the same angles but be different sizes: is not congruent to

13. Same Angles of a Triangle because they are (in this case) the same size Can two triangles of the same angles be congruent? Yes. They could be congruent if they are the same size is congruent to

14. Marking of Congruent Triangles If two triangles are congruent, we often mark corresponding sides and angles like this: is congruent to:

15. Marking of Congruent Triangles The sides marked with one line are equal in length. Similarly for the sides marked with two lines and three lines. The angles marked with one arc are equal in size. Similarly for the angles marked with two arcs and three arcs.

16. How To Find if Triangles are Congruent • Two triangles are congruent if they have: • exactly the same three sides and • exactly the same three angles. • But we don't have to know all three sides and all three angles ...usually three out of the sixis enough. • There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

17. 1. SSS   (side, side, side) If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. SSS stands for "side, side, side“ and means that we have two triangles with all three sides equal. For example: is congruent to:

18. 2. SAS (side, angle, side) If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent. SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal. For example: is congruent to:

19. 3. ASA   (angle, side, angle) If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. ASA stands for "angle, side, angle“ and means that we have two triangles where we know two angles and the included side are equal. For example: is congruent to:

20. 4. AAS   (angle, angle, side) If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. AAS stands for "angle, angle, side“ and means that we have two triangles where we know two angles and the non-included side are equal. For example: is congruent to:

21. 5. HL   (hypotenuse, leg) HL applies only to right angled-triangles! HL stands for "Hypotenuse, Leg" (the longest side of the triangle is called the "hypotenuse", the other two sides are called "legs") and

22. 5. HL   (hypotenuse, leg) If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent. • It means we have two right-angled triangles with • the same length of hypotenuseand • thesame length for one of the other two legs. • It doesn't matter which leg since the triangles could be rotated. • For example: is congruent to 

23. Caution ! Don't Use "AAA" ! Without knowing at least one side, we can't be sure if two triangles are congruent.. AAA means we are given all three angles of a triangle, but no sides. This is not enough information to decide if two triangles are congruent! Because the triangles can have the same angles but be different sizes: For example: is congruent to 

24. Can You Classify the Different Triangles in the Picture Below? Classify the following triangles: AED, ABC, ACD, ACE

25. The Classifications… • Triangle AED = Equilateral, Equiangular • Triangle ABC = Equilateral, Equiangular • Triangle ACD = Isoceles, Obtuse • Triangle ACE = Scalene, Right • So how did you do?