Introduction to Triangles

Can be classified by the angle measures Triangles Can be classified by the angle measures

Has three acute angles (less than 90 degrees) Acute Triangle Has three acute angles (less than 90 degrees)

Triangle with one obtuse angle (greater than 90 degrees) Obtuse Triangle Triangle with one obtuse angle (greater than 90 degrees)

Has one right angle (90 degree) Right Triangle Has one right angle (90 degree)

Can be classified by the number of congruent sides Triangles Can be classified by the number of congruent sides

Has no congruent sides (all angles, and sides are different sizes) Scalene Triangle Has no congruent sides (all angles, and sides are different sizes)

Has at least two congruent sides Isosceles Triangle Has at least two congruent sides least

the angles opposite the congruent sides are also congruent Isosceles Triangle the angles opposite the congruent sides are also congruent

Equilateral Triangle All three sides are congruent Congruent – same size and shape

Which can only happen in a equilateral triangle Equiangular Triangle Triangle with 3 congruent angles Which can only happen in a equilateral triangle

Classifying Triangles

Triangles Cut any shape triangle out of a sheet of paper . Tear off the corners. Piece them together by having the corners touch. The corners form what type of angle?

The sum of the angles of a triangle is 180 degrees Triangles The sum of the angles of a triangle is 180 degrees

Proving sum of the angles of a triangle is 180 degrees Prove A+B + C = 180 G C H 1 2 3 1. Create a line GH that is parallel to AB 2. Label the angles along the straight line, 1, 2, 3 1 3 A B 3. Use what you know about alternate interior angles and label the lower angles in the triangle 4. Since angles 1,2, & 3 create a straight line, we know their sum is 180°. 5. Therefore, we know that that the sum of the internal angles of a triangle always add to 180°

If all the angles must add to 180 and be the same….. Equiangular Triangle Triangle with 3 equal angles If all the angles must add to 180 and be the same….. Then, x+x+x = 180 3x = 180 X = 60

If you know 2 angles, then you can always figure out the 3rd Triangles If you know 2 angles, then you can always figure out the 3rd

Triangle Inequalities

Triangle Inequality Theorem: The sum of two sides of a triangle must be greater than the length of the third side. a + b > c a + c > b b + c > a Example: Determine if it is possible to draw a triangle with side measures 12, 11, and 17. 12 + 11 > 17  Yes 11 + 17 > 12  Yes 12 + 17 > 11  Yes Therefore a triangle can be drawn.

Angle Side Relationship The longest side is across from the largest angle. The shortest side is across from the smallest angle. 54 ° 37 89 B C A BC = 3.2 cm AB = 4.3 cm AC = 5.3 cm

Triangle Inequality – examples… For the triangle, list the angles in order from least to greatest measure. C A B 4 cm 6 cm 5 cm

Exterior Angles B x y w A C An exterior angle formed by a side of the triangle and the extension of another side . In this case, w The remote interior angles the two nonadjacent interior angles. In this case x & y

Find the measure of angle  B The sum of the measure of the angles of a triangle is 1800. Lets call the 3rd internal angle z 400 800 600 z 1200  A C D 40 + 80 + z = 180 120 + z = 180 z = 60 ACB and BCD are supplementary 60 + x = 180  = 120