At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Certainly! Let's complete the proof to show that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^{\circ}\)[/tex].
[tex]\[ \begin{tabular}{|l|l|} \hline Statement & Reason \\ \hline Points $A, B$, and $C$ form a triangle. & Given \\ \hline Let $\overline{DE}$ be a line passing through $B$ and parallel to $\overline{AC}$ & Definition of parallel lines \\ \hline $\angle 3 \cong \angle 5$ and $\angle 1 \cong \angle 4$ & Alternate interior angles are congruent \\ \hline $m\angle 1 = m\angle 4$ and $m\angle 3 = m\angle 5$ & Congruent angles have equal measures \\ \hline $m\angle 4 + m\angle 2 + m\angle 5 = 180^{\circ}$ & Angle addition and definition of a straight line \\ \hline $m\angle 1 + m\angle 2 + m\angle 3 = 180^{\circ}$ & Substitution \\ \hline \end{tabular} \][/tex]
Here’s a detailed explanation of each step:
1. Points [tex]\(A, B, and C\)[/tex] form a triangle.
- This is given.
2. Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex].
- By the definition of parallel lines, we draw a line [tex]\(\overline{DE}\)[/tex] that is parallel to [tex]\(\overline{AC}\)[/tex] and passes through point [tex]\(B\)[/tex].
3. [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex].
- This is because [tex]\(\angle 3\)[/tex] and [tex]\(\angle 5\)[/tex] are alternate interior angles formed by the transversal [tex]\(\overline{BC}\)[/tex] through the parallel lines [tex]\(\overline{DE}\)[/tex] and [tex]\(\overline{AC}\)[/tex]. Similarly, [tex]\(\angle 1\)[/tex] and [tex]\(\angle 4\)[/tex] are alternate interior angles formed by the transversal [tex]\(\overline{AB}\)[/tex] through the parallel lines [tex]\(\overline{DE}\)[/tex] and [tex]\(\overline{AC}\)[/tex]. Alternate interior angles between parallel lines are congruent.
4. [tex]\(m\angle 1 = m\angle 4\)[/tex] and [tex]\(m\angle 3 = m\angle 5\)[/tex].
- Since [tex]\(\angle 1 \cong \angle 4\)[/tex] and [tex]\(\angle 3 \cong \angle 5\)[/tex], their measures are equal.
5. [tex]\(m\angle 4 + m\angle 2 + m\angle 5 = 180^{\circ}\)[/tex].
- As [tex]\(\overline{DE}\)[/tex] forms a straight line, the angles along this straight line add up to [tex]\(180^{\circ}\)[/tex]. Therefore, [tex]\(m\angle 4 + m\angle 2 + m\angle 5 = 180^{\circ}\)[/tex].
6. [tex]\(m\angle 1 + m\angle 2 + m\angle 3 = 180^{\circ}\)[/tex].
- By substituting [tex]\(m\angle 4\)[/tex] with [tex]\(m\angle 1\)[/tex] and [tex]\(m\angle 5\)[/tex] with [tex]\(m\angle 3\)[/tex] from previous equality statements, we get [tex]\(m\angle 1 + m\angle 2 + m\angle 3 = 180^{\circ}\)[/tex].
Thus, the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^{\circ}\)[/tex].
[tex]\[ \begin{tabular}{|l|l|} \hline Statement & Reason \\ \hline Points $A, B$, and $C$ form a triangle. & Given \\ \hline Let $\overline{DE}$ be a line passing through $B$ and parallel to $\overline{AC}$ & Definition of parallel lines \\ \hline $\angle 3 \cong \angle 5$ and $\angle 1 \cong \angle 4$ & Alternate interior angles are congruent \\ \hline $m\angle 1 = m\angle 4$ and $m\angle 3 = m\angle 5$ & Congruent angles have equal measures \\ \hline $m\angle 4 + m\angle 2 + m\angle 5 = 180^{\circ}$ & Angle addition and definition of a straight line \\ \hline $m\angle 1 + m\angle 2 + m\angle 3 = 180^{\circ}$ & Substitution \\ \hline \end{tabular} \][/tex]
Here’s a detailed explanation of each step:
1. Points [tex]\(A, B, and C\)[/tex] form a triangle.
- This is given.
2. Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex].
- By the definition of parallel lines, we draw a line [tex]\(\overline{DE}\)[/tex] that is parallel to [tex]\(\overline{AC}\)[/tex] and passes through point [tex]\(B\)[/tex].
3. [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex].
- This is because [tex]\(\angle 3\)[/tex] and [tex]\(\angle 5\)[/tex] are alternate interior angles formed by the transversal [tex]\(\overline{BC}\)[/tex] through the parallel lines [tex]\(\overline{DE}\)[/tex] and [tex]\(\overline{AC}\)[/tex]. Similarly, [tex]\(\angle 1\)[/tex] and [tex]\(\angle 4\)[/tex] are alternate interior angles formed by the transversal [tex]\(\overline{AB}\)[/tex] through the parallel lines [tex]\(\overline{DE}\)[/tex] and [tex]\(\overline{AC}\)[/tex]. Alternate interior angles between parallel lines are congruent.
4. [tex]\(m\angle 1 = m\angle 4\)[/tex] and [tex]\(m\angle 3 = m\angle 5\)[/tex].
- Since [tex]\(\angle 1 \cong \angle 4\)[/tex] and [tex]\(\angle 3 \cong \angle 5\)[/tex], their measures are equal.
5. [tex]\(m\angle 4 + m\angle 2 + m\angle 5 = 180^{\circ}\)[/tex].
- As [tex]\(\overline{DE}\)[/tex] forms a straight line, the angles along this straight line add up to [tex]\(180^{\circ}\)[/tex]. Therefore, [tex]\(m\angle 4 + m\angle 2 + m\angle 5 = 180^{\circ}\)[/tex].
6. [tex]\(m\angle 1 + m\angle 2 + m\angle 3 = 180^{\circ}\)[/tex].
- By substituting [tex]\(m\angle 4\)[/tex] with [tex]\(m\angle 1\)[/tex] and [tex]\(m\angle 5\)[/tex] with [tex]\(m\angle 3\)[/tex] from previous equality statements, we get [tex]\(m\angle 1 + m\angle 2 + m\angle 3 = 180^{\circ}\)[/tex].
Thus, the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^{\circ}\)[/tex].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
