1. Lesson 3-4 Angles of a Triangle (page 93)
Essential Question
How can you apply parallel lines (planes) to make deductions?

2. Angles of a Triangle A . B.
. C

3. TRIANGLE: . . C ∆ ACB ∆ BAC ∆ BCA ∆ CAB ∆ CBA ∆ ABC A B C three
The figure formed by 3 segments joining 3 noncollinear points.
∆ ABC
Symbol: ____________ A .
three
VERTEX: each of the ______ points. B.
. C
A B C
Vertices: _____ _____ _____
Sides: _____ _____ _____
Angles: _____ _____ _____
∠A ∠B ∠C

4. Classifications of Triangles by Sides
Scalene Triangle
No sides are congruent .

5. Classifications of Triangles by Sides
Isosceles Triangle
At least two sides are congruent .

6. Classifications of Triangles by Sides
Equilateral Triangle
All sides are congruent .

7. Classifications of Triangles by Angles
Acute Triangle
Three acute angles.

8. Classifications of Triangles by Angles
Right Triangle
One right (90º) angle.

9. Classifications of Triangles by Angles
Obtuse Triangle
One obtuse angle.

10. Classifications of Triangles by Angles
Equiangular Triangle
All angles are congruent .

11. Auxiliary Line: A line (ray or segment) added to a
diagram to help in a proof .
Please note that this is to HELP in a proof.
This does not give you license to add lines to every diagram.
There are times when this may be done, but please BEWARE!

12. The sum of the measures of the angles of a triangle is 180º .
Theorem 3-11
The sum of the measures of the angles of a triangle is 180º . B
Given: ∆ ABC
Prove: m∠1 + m∠2 + m∠3 = 180º 2 1 3 A C

13. || - lines ⇒ AIA ≅ See page 94! Substitution PropertyD  B
Given: ∆ ABC
Prove: m∠1 + m∠ 2 + m∠3 = 180º
Proof:
Statements Reasons
Through B draw line BD parallel to line AC Through a point outside a line, there is exactly one line parallel to the given line.
__________________________________________ _____________________________________________ 4 5 2
See page 94! 1 3 A C
m∠ DBC +m∠5 = 180º
m∠ DBC = m∠2 + m∠4
∠ - Addition Postulate
m∠2 + m∠4 + m∠5 = 180º
Substitution Property
|| - lines ⇒ AIA ≅
∠1 ≅ ∠4 OR m∠1 = m∠4
∠3 ≅ ∠5 OR m∠3 = m∠ 5
m∠1 + m∠2 + m∠ 3 = 180º
Substitution Property

14. 50 Example # 1. Find the value of “x”. x + 40 + 90 = 180 x + 130 = 180
40º

15. 45 Example # 2. Find the value of “x”. x + 100 + 35 = 180
100º
35º

16. 55 Example # 3. Find the value of “x”. x + x + 70 = 180 2x + 70 = 180
70º
x º

17. A statement that can be proved easily
COROLLARY:
A statement that can be proved easily
by applying a theorem.

18. Corollary 1
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent . A X Y Z B C
If ∠A ≅ ∠X and ∠B ≅ ∠Y, then ∠C ≅∠Z

19. Each angle of an equiangular triangle has a measure 60º .
Corollary 2
Each angle of an equiangular triangle has a measure 60º .
x º
x º
x º
If 3 x = 180, then x = 60.

20. In a triangle, there can be at most one right angle or obtuse angle.
Corollary 3
In a triangle, there can be at most one right angle or obtuse angle.
Sum = 90º
Sum < 90º
m > 90º

21. The acute angles of a right triangle are complementary .
Corollary 4
The acute angles of a right triangle are complementary .
Sum = 90º
If 2 ∠’s sum = 90º, then they are complementary.

22. Example # 4.
Find the value of “x”.
40º
50º
x º
40º 50
x = _____

23. Example # 5.
Find the value of “x”.
x º
x º
x º 60
x = _____

24. EXTERIOR ANGLE: (of a triangle) the angle
formed when one side of a triangle is extended .
Example: ∠4 4 3
Example: ∠1 & ∠2 2 2 1 1
REMOTE INTERIOR ANGLES: the angles of
the triangle not adjacent to the exterior angle.

25. REMOTE INTERIOR ANGLES:
EXTERIOR ANGLE:
Example: ∠4 1 1 4 3 2 2
REMOTE INTERIOR ANGLES:
Example: ∠1 & ∠2

26. Theorem 3-12
The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.
Given: ∆ ABC
Prove: m∠1 + m∠2 = m∠ 4 B 2 1 3 4 A C

27. Substitution Property
Given: ∆ ABC
Prove: m∠1 + m∠2 = m∠4
Proof:
Statements Reasons
__________________________________________ _____________________________________________ 2
See page 96
C.E. #15! 1 3 4 A C
m∠1 + m∠2+ m∠ 3 = 180º
Sum of m. of∠‘s of ∆ = 180º
m∠3 + m∠ 4 = 180º
∠- Addition Postulate
Substitution Property
m∠1+ m∠2 + m∠3 = m∠3 + m∠4
m∠3 = m∠3
Reflexive Property
m∠1 + m∠ 2 = m∠4
Subtraction Property

28. 40 Example # 6. Find the value of “x”. x + 80 = 120 x = _____ 80º 120º

29. 50 Example # 7. Find the value of “x”. 3x = x + 100 2x = 100 x = _____
100º
3 x º

30. Example # 8.
Find the value of “x”.
x =
60º
150
x = _____
x º

31. Do the Paper Triangle Proofs
Assignment
Written Exercises on pages 97 to 99
RECOMMENDED: 1 to 9 odd numbers
REQUIRED: 10, 11, 13, 15, 17, 18, 19, 20, 25, 26
Do the Paper Triangle Proofs
How can you apply parallel lines (planes) to make deductions?