IDNLearn.com offers a comprehensive solution for all your question and answer needs. Ask anything and receive well-informed answers from our community of experienced professionals.
Sagot :
Let's complete the statements to prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
\begin{tabular}{|l|l|}
\hline Statement & Reason \\
\hline Points [tex]$A, B$[/tex], and [tex]$C$[/tex] form a triangle. & given \\
\hline Let [tex]$\overline{D E}$[/tex] be a line passing through [tex]$B$[/tex] and parallel to [tex]$\overline{A C}$[/tex] & definition of parallel lines \\
\hline [tex]$\angle 3 \cong \angle 5$[/tex] and [tex]$\angle 1 \cong \angle 4$[/tex] & alternate interior angles \\
\hline [tex]$m \angle 1= m \angle 4$[/tex] and [tex]$m \angle 3= m \angle 5$[/tex] & congruent angles have equal measures \\
\hline [tex]$m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}$[/tex] & angle addition and definition of a straight line \\
\hline [tex]$m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}$[/tex] & substitution \\
\hline
\end{tabular}
Here's a detailed, step-by-step explanation of the proof process:
1. Given: Points [tex]\(A, B,\)[/tex] and [tex]\(C\)[/tex] form a triangle.
- Reason: This is the starting information provided in the problem.
2. Definition of parallel lines: Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex].
- Reason: According to the definition, we establish a line [tex]\(\overline{DE}\)[/tex] parallel to [tex]\(\overline{AC}\)[/tex] through point [tex]\(B\)[/tex].
3. Alternate interior angles: [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex].
- Reason: For parallel lines, alternate interior angles are congruent. Since [tex]\(\overline{DE} \parallel \overline{AC}\)[/tex], the angles formed adhere to this property.
4. Congruent angles have equal measures: [tex]\(m \angle 1 = m \angle 4\)[/tex] and [tex]\(m \angle 3 = m \angle 5\)[/tex].
- Reason: When angles are congruent, their measures are equal. Thus, from the congruences in the previous step, we equate their measures.
5. Angle addition and definition of a straight line: [tex]\(m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ\)[/tex].
- Reason: The angles forming a straight line sum to [tex]\(180^\circ\)[/tex]. In this case, the angles along [tex]\(\overline{DE}\)[/tex] (specifically, [tex]\(\angle 4\)[/tex], [tex]\(\angle 2\)[/tex], and [tex]\(\angle 5\)[/tex]) sum up to form a straight line, thus totaling [tex]\(180^\circ\)[/tex].
6. Substitution: [tex]\(m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ\)[/tex].
- Reason: By substituting [tex]\(m \angle 1\)[/tex] for [tex]\(m \angle 4\)[/tex] and [tex]\(m \angle 3\)[/tex] for [tex]\(m \angle 5\)[/tex] from the previously stated equal measures, we conclude that the interior angles of the triangle ([tex]\(\angle 1\)[/tex], [tex]\(\angle 2\)[/tex], and [tex]\(\angle 3\)[/tex]) sum to [tex]\(180^\circ\)[/tex].
Thus, we have proven that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
\begin{tabular}{|l|l|}
\hline Statement & Reason \\
\hline Points [tex]$A, B$[/tex], and [tex]$C$[/tex] form a triangle. & given \\
\hline Let [tex]$\overline{D E}$[/tex] be a line passing through [tex]$B$[/tex] and parallel to [tex]$\overline{A C}$[/tex] & definition of parallel lines \\
\hline [tex]$\angle 3 \cong \angle 5$[/tex] and [tex]$\angle 1 \cong \angle 4$[/tex] & alternate interior angles \\
\hline [tex]$m \angle 1= m \angle 4$[/tex] and [tex]$m \angle 3= m \angle 5$[/tex] & congruent angles have equal measures \\
\hline [tex]$m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}$[/tex] & angle addition and definition of a straight line \\
\hline [tex]$m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}$[/tex] & substitution \\
\hline
\end{tabular}
Here's a detailed, step-by-step explanation of the proof process:
1. Given: Points [tex]\(A, B,\)[/tex] and [tex]\(C\)[/tex] form a triangle.
- Reason: This is the starting information provided in the problem.
2. Definition of parallel lines: Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex].
- Reason: According to the definition, we establish a line [tex]\(\overline{DE}\)[/tex] parallel to [tex]\(\overline{AC}\)[/tex] through point [tex]\(B\)[/tex].
3. Alternate interior angles: [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex].
- Reason: For parallel lines, alternate interior angles are congruent. Since [tex]\(\overline{DE} \parallel \overline{AC}\)[/tex], the angles formed adhere to this property.
4. Congruent angles have equal measures: [tex]\(m \angle 1 = m \angle 4\)[/tex] and [tex]\(m \angle 3 = m \angle 5\)[/tex].
- Reason: When angles are congruent, their measures are equal. Thus, from the congruences in the previous step, we equate their measures.
5. Angle addition and definition of a straight line: [tex]\(m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ\)[/tex].
- Reason: The angles forming a straight line sum to [tex]\(180^\circ\)[/tex]. In this case, the angles along [tex]\(\overline{DE}\)[/tex] (specifically, [tex]\(\angle 4\)[/tex], [tex]\(\angle 2\)[/tex], and [tex]\(\angle 5\)[/tex]) sum up to form a straight line, thus totaling [tex]\(180^\circ\)[/tex].
6. Substitution: [tex]\(m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ\)[/tex].
- Reason: By substituting [tex]\(m \angle 1\)[/tex] for [tex]\(m \angle 4\)[/tex] and [tex]\(m \angle 3\)[/tex] for [tex]\(m \angle 5\)[/tex] from the previously stated equal measures, we conclude that the interior angles of the triangle ([tex]\(\angle 1\)[/tex], [tex]\(\angle 2\)[/tex], and [tex]\(\angle 3\)[/tex]) sum to [tex]\(180^\circ\)[/tex].
Thus, we have proven that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex].
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.