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Transcript 
Parallel Lines and Triangles
Through a point not on a line, there is one
and only one line parallel to the given line.
P
Through a point not on a line, there is one
and only one line parallel to the given line.
P
Given ABC , construct a line through point
B, parallel to AC.
B
A
C
Given ABC , construct a line through point
B, parallel to AC.
P
A
B
R
C
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.
Triangle Angle-Sum Theorem
The sum of the measures of the angles of a
triangle is 180°.
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
B
1 2
Reasons
A
R
3
C
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
1.
ABC , PR || AC
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°
B
1 2
Reasons
A
R
3
C
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
1.
ABC , PR || AC
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°
B
1 2
Reasons
1. Given
A
R
3
C
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
1.
ABC , PR || AC
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°
B
1 2
Reasons
R
3
A
1. Given
2. Angles that form a linear pair are
supplementary.
C
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
1.
ABC , PR || AC
B
1 2
Reasons
R
3
A
C
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°
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
1.
ABC , PR || AC
B
1 2
Reasons
R
3
A
C
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°
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
1.
ABC , PR || AC
B
1 2
Reasons
R
3
A
C
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°
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
1.
ABC , PR || AC
B
1 2
Reasons
R
3
A
C
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°
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
1.
ABC , PR || AC
B
1 2
Reasons
R
3
A
C
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°
Proof of Triangle Angle-Sum Theorem
P
Given: ABC , PR || AC
Prove: mA + m2 + mC = 180°
Statements
1.
ABC , PR || AC
B
1 2
Reasons
R
3
A
C
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)
Exterior Angle of a Triangle
An exterior angle of a triangle is formed by
extending a side of the triangle.
exterior
angle
For each exterior angle, the two nonadjacent
interior angles are called its remote interior
angles.
remote interior angle
exterior
angle
remote interior angle
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
remote interior angle
3
exterior
angle
4
Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2
Statements
Reasons
1
2
3
4
Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2
Statements
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
Reasons
1
2
3
4
Proof of Exterior Angle Theorem
Given: 4 is an exterior angle of the triangle
Prove: m4 = m1 + m2
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
1
2
3
4
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
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
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
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
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
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)
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