1. Parallel Lines and Triangles

2. Through a point not on a line, there is one and only one line parallel to the given line.

3. Through a point not on a line, there is one and only one line parallel to the given line.

4. Given , construct a line through point B, parallel to .

5. Given , construct a line through point B, parallel to .
This auxiliary line (a line added to a diagram to help explain relationships in proofs) can be used to explore the sum of the interior angles in a triangle.

6. Triangle Angle-Sum Theorem
The sum of the measures of the angles of a triangle is 180°.

7. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons

8. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons 1.
2. PBC and 3 are supplementary
3. mPBC + m3 = 180°
4. mPBC = m1 + m2
5. m1 + m2 + m3 = 180°
6. 1  A, 3  C
7. m1 = mA, m3 = mC
8. mA + m2 + mC = 180°

9. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons 1.
1. Given
2. PBC and 3 are supplementary
3. mPBC + m3 = 180°
4. mPBC = m1 + m2
5. m1 + m2 + m3 = 180°
6. 1  A, 3  C
7. m1 = mA, m3 = mC
8. mA + m2 + mC = 180°

10. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons 1.
1. Given
2. PBC and 3 are supplementary
2. Angles that form a linear pair are supplementary.
3. mPBC + m3 = 180°
4. mPBC = m1 + m2
5. m1 + m2 + m3 = 180°
6. 1  A, 3  C
7. m1 = mA, m3 = mC
8. mA + m2 + mC = 180°

11. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons 1.
1. Given
2. PBC and 3 are supplementary
2. Angles that form a linear pair are supplementary.
3. mPBC + m3 = 180°
3. Supplementary angles add to 180°. (2)
4. mPBC = m1 + m2
5. m1 + m2 + m3 = 180°
6. 1  A, 3  C
7. m1 = mA, m3 = mC
8. mA + m2 + mC = 180°

12. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons 1.
1. Given
2. PBC and 3 are supplementary
2. Angles that form a linear pair are supplementary.
3. mPBC + m3 = 180°
3. Supplementary angles add to 180°. (2)
4. mPBC = m1 + m2
4. Angle Addition Postulate
5. m1 + m2 + m3 = 180°
6. 1  A, 3  C
7. m1 = mA, m3 = mC
8. mA + m2 + mC = 180°

13. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons 1.
1. Given
2. PBC and 3 are supplementary
2. Angles that form a linear pair are supplementary.
3. mPBC + m3 = 180°
3. Supplementary angles add to 180°. (2)
4. mPBC = m1 + m2
4. Angle Addition Postulate
5. m1 + m2 + m3 = 180°
5. Substitution Property (3, 4)
6. 1  A, 3  C
7. m1 = mA, m3 = mC
8. mA + m2 + mC = 180°

14. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons 1.
1. Given
2. PBC and 3 are supplementary
2. Angles that form a linear pair are supplementary.
3. mPBC + m3 = 180°
3. Supplementary angles add to 180°. (2)
4. mPBC = m1 + m2
4. Angle Addition Postulate
5. m1 + m2 + m3 = 180°
5. Substitution Property (3, 4)
6. 1  A, 3  C
6. If two parallel lines are cut by a transversal, then alternate interior angles are congruent. (1)
7. m1 = mA, m3 = mC
8. mA + m2 + mC = 180°

15. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons 1.
1. Given
2. PBC and 3 are supplementary
2. Angles that form a linear pair are supplementary.
3. mPBC + m3 = 180°
3. Supplementary angles add to 180°. (2)
4. mPBC = m1 + m2
4. Angle Addition Postulate
5. m1 + m2 + m3 = 180°
5. Substitution Property (3, 4)
6. 1  A, 3  C
6. If two parallel lines are cut by a transversal, then alternate interior angles are congruent. (1)
7. m1 = mA, m3 = mC
7. Congruent angles are = in measure. (6)
8. mA + m2 + mC = 180°

16. Proof of Triangle Angle-Sum TheoremB A C P R 1 2 3
Given:
Prove: mA + m2 + mC = 180°
Statements
Reasons 1.
1. Given
2. PBC and 3 are supplementary
2. Angles that form a linear pair are supplementary.
3. mPBC + m3 = 180°
3. Supplementary angles add to 180°. (2)
4. mPBC = m1 + m2
4. Angle Addition Postulate
5. m1 + m2 + m3 = 180°
5. Substitution Property (3, 4)
6. 1  A, 3  C
6. If two parallel lines are cut by a transversal, then alternate interior angles are congruent. (1)
7. m1 = mA, m3 = mC
7. Congruent angles are = in measure. (6)
8. mA + m2 + mC = 180°
8. Substitution Property (5, 7)

17. Exterior Angle of a Triangle
An exterior angle of a triangle is formed by extending a side of the triangle.
exterior angle

18. For each exterior angle, the two nonadjacent interior angles are called its remote interior angles.

19. Exterior Angle Theorem
The measure of each exterior angle of a triangle equals the sum of its two remote interior angles.
m4 = m1 + m2
remote interior angle 1 2 3 4
exterior angle
remote interior angle

20. Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2 1 2 3 4
Statements
Reasons

21. Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2 1 2 3 4
Statements
Reasons
1. 4 is an exterior angle of the triangle
2. 3 and 4 are supplementary
3. m3 + m4 = 180°
4. m1 + m2 + m3 = 180°
5. m3 + m4 = m1 + m2 + m3
6. m3 = m3
7. m4 = m1 + m2

22. Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2 1 2 3 4
Statements
Reasons
1. 4 is an exterior angle of the triangle
1. Given
2. 3 and 4 are supplementary
3. m3 + m4 = 180°
4. m1 + m2 + m3 = 180°
5. m3 + m4 = m1 + m2 + m3
6. m3 = m3
7. m4 = m1 + m2

23. Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2 1 2 3 4
Statements
Reasons
1. 4 is an exterior angle of the triangle
1. Given
2. 3 and 4 are supplementary
2. Angles that form a linear pair are supplementary.
3. m3 + m4 = 180°
4. m1 + m2 + m3 = 180°
5. m3 + m4 = m1 + m2 + m3
6. m3 = m3
7. m4 = m1 + m2

24. Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2 1 2 3 4
Statements
Reasons
1. 4 is an exterior angle of the triangle
1. Given
2. 3 and 4 are supplementary
2. Angles that form a linear pair are supplementary.
3. m3 + m4 = 180°
3. Supplementary angles add to 180°. (2)
4. m1 + m2 + m3 = 180°
5. m3 + m4 = m1 + m2 + m3
6. m3 = m3
7. m4 = m1 + m2

25. Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2 1 2 3 4
Statements
Reasons
1. 4 is an exterior angle of the triangle
1. Given
2. 3 and 4 are supplementary
2. Angles that form a linear pair are supplementary.
3. m3 + m4 = 180°
3. Supplementary angles add to 180°. (2)
4. m1 + m2 + m3 = 180°
4. The sum of the interior angles of a triangle is 180°.
5. m3 + m4 = m1 + m2 + m3
6. m3 = m3
7. m4 = m1 + m2

26. Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2 1 2 3 4
Statements
Reasons
1. 4 is an exterior angle of the triangle
1. Given
2. 3 and 4 are supplementary
2. Angles that form a linear pair are supplementary.
3. m3 + m4 = 180°
3. Supplementary angles add to 180°. (2)
4. m1 + m2 + m3 = 180°
4. The sum of the interior angles of a triangle is 180°.
5. m3 + m4 = m1 + m2 + m3
5. Substitution Property (3, 4)
6. m3 = m3
7. m4 = m1 + m2

27. Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2 1 2 3 4
Statements
Reasons
1. 4 is an exterior angle of the triangle
1. Given
2. 3 and 4 are supplementary
2. Angles that form a linear pair are supplementary.
3. m3 + m4 = 180°
3. Supplementary angles add to 180°. (2)
4. m1 + m2 + m3 = 180°
4. The sum of the interior angles of a triangle is 180°.
5. m3 + m4 = m1 + m2 + m3
5. Substitution Property (3, 4)
6. m3 = m3
6. Reflexive Property
7. m4 = m1 + m2

28. Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2 1 2 3 4
Statements
Reasons
1. 4 is an exterior angle of the triangle
1. Given
2. 3 and 4 are supplementary
2. Angles that form a linear pair are supplementary.
3. m3 + m4 = 180°
3. Supplementary angles add to 180°. (2)
4. m1 + m2 + m3 = 180°
4. The sum of the interior angles of a triangle is 180°.
5. m3 + m4 = m1 + m2 + m3
5. Substitution Property (3, 4)
6. m3 = m3
6. Reflexive Property
7. m4 = m1 + m2
7. Subtraction Property (5, 7)