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Let's prove that the sum of the interior angle measures of a triangle is \(180^\circ\) using the parallel lines \(y \parallel z\).
Given: \(y \parallel z\)
Prove: \(m \angle 5 + m \angle 2 + m \angle 0 = 180^\circ\)
### Proof:
1. Statements: \(y \parallel z\)
Reasons: Given
2. Statements: \(\angle 5 = \angle 1\)
Reasons: Alternate interior angles are congruent when two parallel lines are cut by a transversal.
3. Statements: \(\angle 6 = \angle 2\)
Reasons: Alternate interior angles are congruent when two parallel lines are cut by a transversal.
4. Statements: \(\angle 0\) is an exterior angle of the triangle.
Reasons: Definition of an exterior angle.
5. Statements: \(\angle 0 = \angle 1 + \angle 2\)
Reasons: The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent interior angles.
6. Statements: \(m \angle 5 + m \angle 2 + m \angle 0\)
Reasons: Summation of the angles as per the question.
7. Statements: Substitute \(\angle 5\) and \(\angle 0\):
[tex]\[ m \angle 5 + m \angle 2 + m \angle 0 = m \angle 1 + m \angle 2 + (m \angle 1 + m \angle 2) \][/tex]
Reasons: Because \(\angle 5 = \angle 1\) and \(\angle 0 = \angle 1 + \angle 2\).
8. Statements: Combine like terms:
[tex]\[ m \angle 1 + m \angle 2 + m \angle 1 + m \angle 2 = 2m \angle 1 + 2m \angle 2 \][/tex]
Reasons: Simplification
9. Statements: The sum of the interior angles in a triangle is always \(180^\circ\).
Reasons: Triangle Angle Sum Theorem
10. Statements: Therefore, \(2m \angle 1 + 2m \angle 2 = 180^\circ\).
Reasons: From the previous steps and the established fact that the sum of the interior angles of a triangle is \(180^\circ\).
Thus, we have proved that the sum of the interior angle measures of a triangle is \(180^\circ\).
\begin{tabular}{|c|l|}
\hline
\textbf{Statements} & \textbf{Reasons} \\
\hline
1. \(y \parallel z\) & 1. Given \\
2. \(\angle 5 = \angle 1\) & 2. Alternate interior angles are congruent. \\
3. \(\angle 6 = \angle 2\) & 3. Alternate interior angles are congruent. \\
4. \(\angle 0\) is an exterior angle. & 4. Definition of an exterior angle. \\
5. \(\angle 0 = \angle 1 + \angle 2\) & 5. Exterior Angle Theorem. \\
6. \(m \angle 5 + m \angle 2 + m \angle 0\) & 6. Summation of angles required. \\
7. \(m \angle 5 + m \angle 2 + m \angle 0 = \angle 1 + \angle 2 + (\angle 1 + \angle 2)\) & 7. Substitution with \(\angle 5 = \angle 1\) and \(\angle 0 = \angle 1 + \angle 2\). \\
8. \(2m \angle 1 + 2m \angle 2\) & 8. Combining like terms. \\
9. The sum of the angles in a triangle is \(180^\circ\). & 9. Triangle Angle Sum Theorem \\
10. \(2m \angle 1 + 2m \angle 2 = 180^\circ\). & 10. Conclusion from previous steps. \\
\hline
\end{tabular}
Hence, we have completed the proof.
Given: \(y \parallel z\)
Prove: \(m \angle 5 + m \angle 2 + m \angle 0 = 180^\circ\)
### Proof:
1. Statements: \(y \parallel z\)
Reasons: Given
2. Statements: \(\angle 5 = \angle 1\)
Reasons: Alternate interior angles are congruent when two parallel lines are cut by a transversal.
3. Statements: \(\angle 6 = \angle 2\)
Reasons: Alternate interior angles are congruent when two parallel lines are cut by a transversal.
4. Statements: \(\angle 0\) is an exterior angle of the triangle.
Reasons: Definition of an exterior angle.
5. Statements: \(\angle 0 = \angle 1 + \angle 2\)
Reasons: The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent interior angles.
6. Statements: \(m \angle 5 + m \angle 2 + m \angle 0\)
Reasons: Summation of the angles as per the question.
7. Statements: Substitute \(\angle 5\) and \(\angle 0\):
[tex]\[ m \angle 5 + m \angle 2 + m \angle 0 = m \angle 1 + m \angle 2 + (m \angle 1 + m \angle 2) \][/tex]
Reasons: Because \(\angle 5 = \angle 1\) and \(\angle 0 = \angle 1 + \angle 2\).
8. Statements: Combine like terms:
[tex]\[ m \angle 1 + m \angle 2 + m \angle 1 + m \angle 2 = 2m \angle 1 + 2m \angle 2 \][/tex]
Reasons: Simplification
9. Statements: The sum of the interior angles in a triangle is always \(180^\circ\).
Reasons: Triangle Angle Sum Theorem
10. Statements: Therefore, \(2m \angle 1 + 2m \angle 2 = 180^\circ\).
Reasons: From the previous steps and the established fact that the sum of the interior angles of a triangle is \(180^\circ\).
Thus, we have proved that the sum of the interior angle measures of a triangle is \(180^\circ\).
\begin{tabular}{|c|l|}
\hline
\textbf{Statements} & \textbf{Reasons} \\
\hline
1. \(y \parallel z\) & 1. Given \\
2. \(\angle 5 = \angle 1\) & 2. Alternate interior angles are congruent. \\
3. \(\angle 6 = \angle 2\) & 3. Alternate interior angles are congruent. \\
4. \(\angle 0\) is an exterior angle. & 4. Definition of an exterior angle. \\
5. \(\angle 0 = \angle 1 + \angle 2\) & 5. Exterior Angle Theorem. \\
6. \(m \angle 5 + m \angle 2 + m \angle 0\) & 6. Summation of angles required. \\
7. \(m \angle 5 + m \angle 2 + m \angle 0 = \angle 1 + \angle 2 + (\angle 1 + \angle 2)\) & 7. Substitution with \(\angle 5 = \angle 1\) and \(\angle 0 = \angle 1 + \angle 2\). \\
8. \(2m \angle 1 + 2m \angle 2\) & 8. Combining like terms. \\
9. The sum of the angles in a triangle is \(180^\circ\). & 9. Triangle Angle Sum Theorem \\
10. \(2m \angle 1 + 2m \angle 2 = 180^\circ\). & 10. Conclusion from previous steps. \\
\hline
\end{tabular}
Hence, we have completed the proof.
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