Join IDNLearn.com and become part of a knowledge-sharing community that thrives on curiosity. Ask anything and get well-informed, reliable answers from our knowledgeable community members.
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 every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.
