This document defines key vocabulary terms related to triangles, including different types of triangles based on angle measures. It presents the Triangle Angle-Sum Theorem and Exterior Angle Theorem, along with their corollaries. It includes three multi-step examples solving for angle measures using these theorems and properties of triangles. The document provides definitions, theorems, examples and step-by-step workings to explain angles of triangles.

1. SECTION 4-1
Angles ofTriangles

2. ESSENTIAL QUESTIONS
• How do you apply theTriangle Angle-SumTheorem?
• How do you apply the Exterior AngleTheorem?

3. VOCABULARY
1.AcuteTriangle:
2. EquiangularTriangle:
3. ObtuseTriangle:
4. RightTriangle:

4. VOCABULARY
1.AcuteTriangle: A triangle in which all three angles
have a measure of less than 90 degrees
2. EquiangularTriangle:
3. ObtuseTriangle:
4. RightTriangle:

5. VOCABULARY
1.AcuteTriangle: A triangle in which all three angles
have a measure of less than 90 degrees
2. EquiangularTriangle: A triangle in which all three
angles have a measure of 60 degrees, thus making
them all equal
3. ObtuseTriangle:
4. RightTriangle:

6. VOCABULARY
1.AcuteTriangle: A triangle in which all three angles
have a measure of less than 90 degrees
2. EquiangularTriangle: A triangle in which all three
angles have a measure of 60 degrees, thus making
them all equal
3. ObtuseTriangle: A triangle in which one of the angles
has a measure greater than 90 degrees
4. RightTriangle:

7. VOCABULARY
1.AcuteTriangle: A triangle in which all three angles
have a measure of less than 90 degrees
2. EquiangularTriangle: A triangle in which all three
angles have a measure of 60 degrees, thus making
them all equal
3. ObtuseTriangle: A triangle in which one of the angles
has a measure greater than 90 degrees
4. RightTriangle:A triangle in which one of the angles
has a measure of 90 degrees

8. VOCABULARY
6. IsoscelesTriangle:
7. ScaleneTriangle:
5. EquilateralTriangle:

9. VOCABULARY
6. IsoscelesTriangle:
7. ScaleneTriangle:
5. EquilateralTriangle: A triangle in which all three sides
have the same measure

10. VOCABULARY
6. IsoscelesTriangle: A triangle in which at least two sides
have the same measure
7. ScaleneTriangle:
5. EquilateralTriangle: A triangle in which all three sides
have the same measure

11. VOCABULARY
6. IsoscelesTriangle: A triangle in which at least two sides
have the same measure
7. ScaleneTriangle: A triangle in which no two sides have
the same measure
5. EquilateralTriangle: A triangle in which all three sides
have the same measure

12. VOCABULARY
8.Auxiliary Line:
9. Exterior Angle:
10. Remote Interior Angles:
11. Flow Proof:

13. VOCABULARY
8.Auxiliary Line: An extra line or segment that is added to
a figure to help analyze geometric relationships
9. Exterior Angle:
10. Remote Interior Angles:
11. Flow Proof:

14. VOCABULARY
8.Auxiliary Line: An extra line or segment that is added to
a figure to help analyze geometric relationships
9. Exterior Angle: Formed outside a triangle when one side
of the triangle is extended;The exterior angle is adjacent
to the interior angle of the triangle
10. Remote Interior Angles:
11. Flow Proof:

15. VOCABULARY
8.Auxiliary Line: An extra line or segment that is added to
a figure to help analyze geometric relationships
9. Exterior Angle: Formed outside a triangle when one side
of the triangle is extended;The exterior angle is adjacent
to the interior angle of the triangle
10. Remote Interior Angles: The two interior angles that
are not adjacent to a given exterior angle
11. Flow Proof:

16. VOCABULARY
8.Auxiliary Line: An extra line or segment that is added to
a figure to help analyze geometric relationships
9. Exterior Angle: Formed outside a triangle when one side
of the triangle is extended;The exterior angle is adjacent
to the interior angle of the triangle
10. Remote Interior Angles: The two interior angles that
are not adjacent to a given exterior angle
11. Flow Proof: Uses statements written in boxes with
arrows to show a logical progression of an argument

17. THEOREMS & COROLLARIES
4.1 -Triangle Angle-SumTheorem:
4.2 - Exterior AngleTheorem:
4.1 Corollary:
4.2 Corollary:

18. THEOREMS & COROLLARIES
4.1 -Triangle Angle-SumTheorem: The sum of the
measures of the angles of any triangle is 180°
4.2 - Exterior AngleTheorem:
4.1 Corollary:
4.2 Corollary:

19. THEOREMS & COROLLARIES
4.1 -Triangle Angle-SumTheorem: The sum of the
measures of the angles of any triangle is 180°
4.2 - Exterior AngleTheorem: The measure of an exterior
angle of a triangle is equal to the sum of the measures of
the two remote interior angles
4.1 Corollary:
4.2 Corollary:

20. THEOREMS & COROLLARIES
4.1 -Triangle Angle-SumTheorem: The sum of the
measures of the angles of any triangle is 180°
4.2 - Exterior AngleTheorem: The measure of an exterior
angle of a triangle is equal to the sum of the measures of
the two remote interior angles
4.1 Corollary: The acute angles of a right triangle are
complementary
4.2 Corollary:

21. THEOREMS & COROLLARIES
4.1 -Triangle Angle-SumTheorem: The sum of the
measures of the angles of any triangle is 180°
4.2 - Exterior AngleTheorem: The measure of an exterior
angle of a triangle is equal to the sum of the measures of
the two remote interior angles
4.1 Corollary: The acute angles of a right triangle are
complementary
4.2 Corollary: There can be at most one right or obtuse
angle in a triangle

22. EXAMPLE 1
The diagram shows the paths a ball is thrown in a game
played by kids. Find the measure of each numbered angle.

23. EXAMPLE 1
The diagram shows the paths a ball is thrown in a game
played by kids. Find the measure of each numbered angle.
m∠1=180 − 43− 74

24. EXAMPLE 1
The diagram shows the paths a ball is thrown in a game
played by kids. Find the measure of each numbered angle.
m∠1=180 − 43− 74 = 63°

25. EXAMPLE 1
The diagram shows the paths a ball is thrown in a game
played by kids. Find the measure of each numbered angle.
m∠1=180 − 43− 74 = 63°
m∠2

26. EXAMPLE 1
The diagram shows the paths a ball is thrown in a game
played by kids. Find the measure of each numbered angle.
m∠1=180 − 43− 74 = 63°
m∠2 = 63°

27. EXAMPLE 1
The diagram shows the paths a ball is thrown in a game
played by kids. Find the measure of each numbered angle.
m∠1=180 − 43− 74 = 63°
m∠2 = 63°
m∠3 =180 − 63− 79

28. EXAMPLE 1
The diagram shows the paths a ball is thrown in a game
played by kids. Find the measure of each numbered angle.
m∠1=180 − 43− 74 = 63°
m∠2 = 63°
m∠3 =180 − 63− 79 = 38°

29. EXAMPLE 2
Find m∠FLW.

30. EXAMPLE 2
Find m∠FLW.
m∠FLW = m∠LOW + m∠OWL

31. EXAMPLE 2
Find m∠FLW.
m∠FLW = m∠LOW + m∠OWL
2x − 48 = x + 32

32. EXAMPLE 2
Find m∠FLW.
m∠FLW = m∠LOW + m∠OWL
2x − 48 = x + 32
x − 48 = 32

33. EXAMPLE 2
Find m∠FLW.
m∠FLW = m∠LOW + m∠OWL
2x − 48 = x + 32
x − 48 = 32
x = 80

34. EXAMPLE 2
Find m∠FLW.
m∠FLW = m∠LOW + m∠OWL
2x − 48 = x + 32
x − 48 = 32
x = 80
m∠FLW = 2(80)− 48

35. EXAMPLE 2
Find m∠FLW.
m∠FLW = m∠LOW + m∠OWL
2x − 48 = x + 32
x − 48 = 32
x = 80
m∠FLW = 2(80)− 48 =160 − 48

36. EXAMPLE 2
Find m∠FLW.
m∠FLW = m∠LOW + m∠OWL
2x − 48 = x + 32
x − 48 = 32
x = 80
m∠FLW = 2(80)− 48 =160 − 48 =112°

37. EXAMPLE 3
Find the measure of each numbered angle.

38. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41

39. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°

40. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48

41. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°

42. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42

43. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42 = 48°

44. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42 = 48°
90 − 34

45. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42 = 48°
90 − 34 = 56

46. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42 = 48°
90 − 34 = 56
56°

47. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42 = 48°
90 − 34 = 56
56°
m∠2 =180 − 56 − 48

48. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42 = 48°
90 − 34 = 56
56°
m∠2 =180 − 56 − 48 = 76°

49. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42 = 48°
90 − 34 = 56
56°
m∠2 =180 − 56 − 48 = 76°
m∠1=180 − 76

50. EXAMPLE 3
Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42 = 48°
90 − 34 = 56
56°
m∠2 =180 − 56 − 48 = 76°
m∠1=180 − 76 =104°