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

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

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.

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

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

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

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

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

Congruent Triangles If two triangles are congruent they will have exactly the same three sides and exactly the same three angles. If two triangles are congruent they will have exactly the same three sides and 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. 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.

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

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

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

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

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

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 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. The angles marked with one arc are equal in size. Similarly for the angles marked with two arcs and three arcs.

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

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

2. SAS (side, angle, side) SAS stands for "side, angle, side" 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: For example: is congruent to: is congruent to: 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.

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

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

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

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

Caution ! Don't Use "AAA" ! AAA means we are given all three 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! This is not enough information to decide if two triangles are congruent! Because the triangles can have the same angles but be different sizes: Because the triangles can have the same angles but be different sizes: For example: For example: is congruent to is congruent to Without knowing at least one side, we can't be sure if two triangles are congruent..

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

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