1. Unit 3
Triangles

2. Classifying Triangles
Lesson 3.1
Classifying Triangles

3. Lesson 3.1 Objectives
Classify triangles according to their side lengths. (G1.2.1)
Classify triangles according to their angle measures. (G1.2.1)
Find a missing angle using the Triangle Sum Theorem. (G1.2.2)
Find a missing angle using the Exterior Angle Theorem. (G1.2.2)

4. Classification of Triangles by Sides
Equilateral
Isosceles
Scalene
Looks Like
Characteristics
3 congruent sides
2 congruent sides
No Congruent Sides

5. Classification of Triangles by Angles
Name
Acute
Equiangular
Right
Obtuse
Looks Like
Characteristics
ALL acute angles
ALL congruent angles
ONLY 1 right angles
ONLY 1 obtuse angle

6. Example 3.1
Classify the following triangles by their sides and their angles.
Scalene
Obtuse
Scalene
Right
Isosceles
Acute
Equilateral
Equiangular

7. Vertex
The vertex of a triangle is any point at which two sides are joined.
It is a corner of a triangle.
There are 3 in every triangle

8. How to Name a Triangle
To name a triangle, simply draw a small triangle followed by its vertices.
We usually try to name the vertices in alphabetical order, when possible.
Example:
ABC

9. More Parts of Triangles
If you were to extend the sides you will see that more angles would be formed.
So we need to keep them separate
There are three angles called interior angles because they are inside the triangle.
There are three new angles called exterior angles because they lie outside the triangle.

10. Theorem 4.1: Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180o.
mA + mB + mC = 180o

11. Example 3.2
Solve for x and then classify the triangle based on its angles.
Acute
75o
50o
3x + 2x + 55 = 180
Triangle Sum Theorem
5x + 55 = 180
Simplify
5x = 125
SPOE
x = 25
DPOE

12. Example 3.3
Solve for x and classify each triangle by angle measure.
Right
Acute

13. Example 3.4
Draw a sketch of the triangle described. Mark the triangle with symbols to indicate the necessary information.
Acute Isosceles
Equilateral
Right Scalene

14. Example 3.5
Draw a sketch of the triangle described. Mark the triangle with specific angle measures, side lengths, or symbols to indicate the necessary information.
Obtuse Scalene
Right Isosceles
Right Equilateral
(Not Possible)

15. Theorem 4.2: Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

16. Example 3.6 Solve for x Exterior Angles Theorem Combine Like Terms
Subtraction Property
Addition Property
Division Property

17. Corollary to the Triangle Sum Theorem
A corollary to a theorem is a statement that can be proved easily using the original theorem itself.
This is treated just like a theorem or a postulate in proofs.
The acute angles in a right triangle are complementary.

18. Example 3.7 Find the unknown angle measures. VA VA
If you don’t like the Exterior Angle Theorem, then find m2 first using the Linear Pair Postulate.
Then find m1 using the Angle Sum Theorem. VA VA

19. Homework 3.1
Lesson 3.1 – All Sections
p1-6
Due Tomorrow

20. Inequalities in One Triangle
Lesson 3.2
Inequalities in One Triangle

21. Lesson 3.2 Objectives
Order the angles in a triangle from smallest to largest based on given side lengths. (G1.2.2)
Order the side lengths of a triangle from smallest to largest based on given angle measures. (G1.2.2)

22. Theorem 5.10: Side Lengths of a Triangle Theorem
If two sides of a triangle unequal, then the measures of the angles opposite theses sides are also unequal, with the greater angle being opposite the greater side.
Basically, the largest angle is found opposite the largest side.
Basically, the largest side is found opposite the largest angle.
2nd Largest Angle
Longest side
Smallest Side
Smallest Angle
Largest Angle
2nd Longest Side

23. Theorem 5.11: Angle Measures of a Triangle Theorem
If two angles of a triangle unequal, then the measures of the sides opposite theses angles are also unequal, with the greater side being opposite the greater angle.
Basically, the largest angle is found opposite the largest side.
Basically, the largest side is found opposite the largest angle.
2nd Largest Angle
Longest side
Smallest Side
Smallest Angle
Largest Angle
2nd Longest Side

24. Example 3.8
Order the angles from largest to smallest.

25. Example 3.9
Order the sides from largest to smallest.
33o

26. Example 3.10 Order the angles from largest to smallest.
In ABC AB = 12 BC = 11 AC = 5.8
Order the sides from largest to smallest.
In XYZ mX = 25o mY = 33o mZ = 122o

27. Homework 3.2 Lesson 3.2 – Inequalities in One Triangle Due Tomorrow
p7-8
Due Tomorrow
Quiz Friday, October 15th

28. Isosceles and Equilateral Triangles
Lesson 3.3
Isosceles and
Equilateral Triangles

29. Lesson 3.3 Objectives
Utilize the Base Angles Theorem to solve for angle measures. (G1.2.2)
Utilize the Converse of the Base Angles Theorem to solve for side lengths. (G1.2.2)
Identify properties of equilateral triangles to solve for side lengths and angle measures. (G1.2.2)

30. Special Parts of an Isosceles Triangle
An isosceles triangle has only two congruent sides
Those two congruent sides are called legs.
The third side is called the base.
legs
base

31. Isosceles Triangle Theorems
Theorem 4.6: Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent to each other.
Theorem 4.7: Converse of Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.

32. Example 3.11 Solve for x and y. + = 90o = 90o = 45 45 = 45o 55o 55o

33. Equilateral Triangles
Corollary to Theorem 4.6
If a triangle is equilateral, then it is equiangular.
Corollary to Theorem 4.7
If a triangle is equiangular, then it is equilateral.

34. Example 3.12 Solve for x and y. 5xo 5xo
It does not matter which two sides you set equal to each other, just pick the pair that looks the easiest!
Or…In order for a triangle to be equiangular, all angles must equal…

35. Homework 3.3 Lesson 3.3 – Isosceles and Equilateral Triangles
p9-11
Due Tomorrow
Quiz Tomorrow
Tuesday, October 19th

36. Medians And Altitudes of Triangles
Lesson 3.4
Medians
And
Altitudes of Triangles

37. Lesson 3.4 Objectives
Identify a median, an altitude, and a perpendicular bisector of a triangle. (G1.2.5)
Identify a centroid of a triangle.
Utilize medians and altitudes to solve for missing parts of a triangle. (G1.2.5)
Identify the orthocenter of a triangle.

38. Perpendicular Bisector
A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called the perpendicular bisector.

39. Triangle Medians
A median of a triangle is a segment that does the following:
Contains one endpoint at a vertex of the triangle, and
Contains its other endpoint at the midpoint of the opposite side of the triangle. A B C D

40. Centroid
Remember: All medians intersect the midpoint of the opposite side.
When all three medians are drawn in, they intersect to form the centroid of a triangle.
This forms a point of concurrency which is defined as a point formed by the intersection of two or more lines.
The centroid happens to find the balance point for any triangle.
In Physics, this is how we locate the center of mass.
Obtuse
Acute
Right

41. Theorem 5.7: Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side.
The centroid is 2/3 the distance from any vertex to the opposite side.
Or said another way, the centroid is twice as far away from the opposite angle as it is to the nearest side.
AP = 2/3AE
BP = 2/3BF
CP = 2/3CD

42. Example 3.13
S is the centroid of RTW, RS = 4, VW = 6, and TV = 9. Find the following: RV 6 SU 2
Half of 4 is 2 RU
4 + 2 = 6 RW 12 TS
6 is 2/3 of 9 SV 3
Half of 6, which is the other part of the median.

43. Altitudes
An altitude of a triangle is the perpendicular segment from a vertex to the opposite side.
It does not bisect the angle.
It does not bisect the side.
The altitude is often thought of as the height.
While true, there are 3 altitudes in every triangle but only 1 height!

44. Orthocenter
The three altitudes of a triangle intersect at a point that we call the orthocenter of the triangle.
The orthocenter can be located:
inside the triangle
outside the triangle, or
on one side of the triangle
Obtuse
Right
Acute
The orthocenter of a right triangle will always be located at the vertex that forms the right angle.

45. Example 3.14
Is segment BD a median, altitude, or perpendicular bisector of ABC?
Hint: It could be more than one!
Perpendicular
Bisector
Altitude
Median
Median
None
None

46. Homework 3.4
Lesson 3.4 – Altitudes and Medians
p12-13
Due Tomorrow

47. Area and Perimeter of Triangles
Lesson 3.5
Area
and
Perimeter of Triangles

48. Lesson 3.5 Objectives
Find the perimeter and area of triangles. (G1.2.2)

49. Reviewing Altitudes
Determine the size of the altitudes of the following triangles. 6 16
If it is a right triangle, then you can use Pythagorean Theorem to solve for the missing side length. ?

50. Area
The area of a figure is defined as “the amount of space inside the boundary of a flat (2-dimensional) object”
Because of the 2-dimensional nature, the units to measure area will always be “squared.”
For example:
in2 or square inches
ft2 or square feet
m2 or square meters
mi2 or square miles
The area of a rectangle has up until now been found by taking:
length x width (l x w)
We will now change the wording slightly to fit a more general pattern for all shapes, and that is:
base x height (b x h)
That general pattern will exist as long as the base and height form a right angle.
Or said another way, the base and height both touch the right angle. l b h w

51. Area of a Triangle
The area of a triangle is found by taking one-half the base times the height of the triangle.
Again the base and height form a right angle.
Notice that the base is an actual side of the triangle, and…
The height is nothing more than the altitude of the triangle drawn from the base to the opposite vertex. b h h b

52. Perimeter of a Triangle
The perimeter of a triangle is found by taking the sum of all three sides of the triangle.
So basically you need to add all three sides together.
The perimeter is a 1-dimensional measurement, so the units should not have an exponent on them.
Example: in ft m mi b a h c

53. Example 3.15
Find the area and perimeter of the following triangles.

54. Homework 3.5 Lesson 3.5 – Area and Perimeter of Triangles Due Tomorrow