Answer :
Certainly! Let's complete the proof step-by-step to show that the sum of a triangle's interior angle measures is [tex]\(180^\circ\)[/tex].
### Statements and Reasons
1. Statement: Construct a line through [tex]\(B\)[/tex] parallel to [tex]\(\overline{AC}\)[/tex].
Reason: Given a line, we can construct a parallel line through a point not on the given line.
2. Statement: [tex]\( m/\angle A = \)[/tex] alternate interior angles formed by parallel lines have equal measures.
Reason: Alternate interior angles formed by parallel lines have equal measures.
3. Statement: [tex]\( m/\angle B = 65^\circ \)[/tex]
Reason: Given.
4. Statement: [tex]\( m/\angle C = \)[/tex] alternate interior angles formed by parallel lines have equal measures.
Reason: Alternate interior angles formed by parallel lines have equal measures.
5. Statement: [tex]\( m/\angle A + m/\angle B + m/\angle C = 180^\circ \)[/tex]
Reason: Substitution and addition.
Let’s go through each step in more detail.
1. Construct a line through [tex]\(B\)[/tex] parallel to [tex]\(\overline{AC}\)[/tex]: Imagine your triangle [tex]\(\triangle ABC\)[/tex] with vertices [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]. Draw a line through point [tex]\(B\)[/tex] that is parallel to the side [tex]\(\overline{AC}\)[/tex]. This new line helps us apply the properties of parallel lines and transversals.
2. [tex]\( m/\angle A = \)[/tex] alternate interior angles formed by parallel lines have equal measures:
Because we drew a parallel line through [tex]\(B\)[/tex], the angle at vertex [tex]\(A\)[/tex] of the triangle [tex]\(\angle A\)[/tex] corresponds to an alternate interior angle created by the transversal [tex]\(\overline{AB}\)[/tex] and the parallel lines. Thus, [tex]\( m/\angle A\)[/tex] is equal to the measure of the alternate interior angle.
3. [tex]\( m/\angle B = 65^\circ \)[/tex]: This is a given in the problem.
4. [tex]\( m/\angle C = \)[/tex] alternate interior angles formed by parallel lines have equal measures: Similar to step 2, the angle at vertex [tex]\(C\)[/tex], [tex]\(\angle C\)[/tex], also corresponds to an alternate interior angle created by the transversal [tex]\(\overline{BC}\)[/tex] and the parallel lines. Thus, [tex]\( m /\angle C\)[/tex] is equal to the measure of the alternate interior angle.
5. [tex]\( m/\angle A + m/\angle B + m/\angle C = 180^\circ \)[/tex]:
By substituting the measures from steps 2, 3, and 4 into this equation, and using the property that alternate interior angles are equal, we find this sum to be [tex]\(180^\circ\)[/tex].
Therefore, the proof shows that the sum of the interior angles in a triangle is always [tex]\(180^\circ\)[/tex] as this conclusion is derived using properties that hold for any triangle and parallel lines, not just specific measurements or configurations. This means we have proven it true for:
(A) Every triangle.
### Statements and Reasons
1. Statement: Construct a line through [tex]\(B\)[/tex] parallel to [tex]\(\overline{AC}\)[/tex].
Reason: Given a line, we can construct a parallel line through a point not on the given line.
2. Statement: [tex]\( m/\angle A = \)[/tex] alternate interior angles formed by parallel lines have equal measures.
Reason: Alternate interior angles formed by parallel lines have equal measures.
3. Statement: [tex]\( m/\angle B = 65^\circ \)[/tex]
Reason: Given.
4. Statement: [tex]\( m/\angle C = \)[/tex] alternate interior angles formed by parallel lines have equal measures.
Reason: Alternate interior angles formed by parallel lines have equal measures.
5. Statement: [tex]\( m/\angle A + m/\angle B + m/\angle C = 180^\circ \)[/tex]
Reason: Substitution and addition.
Let’s go through each step in more detail.
1. Construct a line through [tex]\(B\)[/tex] parallel to [tex]\(\overline{AC}\)[/tex]: Imagine your triangle [tex]\(\triangle ABC\)[/tex] with vertices [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]. Draw a line through point [tex]\(B\)[/tex] that is parallel to the side [tex]\(\overline{AC}\)[/tex]. This new line helps us apply the properties of parallel lines and transversals.
2. [tex]\( m/\angle A = \)[/tex] alternate interior angles formed by parallel lines have equal measures:
Because we drew a parallel line through [tex]\(B\)[/tex], the angle at vertex [tex]\(A\)[/tex] of the triangle [tex]\(\angle A\)[/tex] corresponds to an alternate interior angle created by the transversal [tex]\(\overline{AB}\)[/tex] and the parallel lines. Thus, [tex]\( m/\angle A\)[/tex] is equal to the measure of the alternate interior angle.
3. [tex]\( m/\angle B = 65^\circ \)[/tex]: This is a given in the problem.
4. [tex]\( m/\angle C = \)[/tex] alternate interior angles formed by parallel lines have equal measures: Similar to step 2, the angle at vertex [tex]\(C\)[/tex], [tex]\(\angle C\)[/tex], also corresponds to an alternate interior angle created by the transversal [tex]\(\overline{BC}\)[/tex] and the parallel lines. Thus, [tex]\( m /\angle C\)[/tex] is equal to the measure of the alternate interior angle.
5. [tex]\( m/\angle A + m/\angle B + m/\angle C = 180^\circ \)[/tex]:
By substituting the measures from steps 2, 3, and 4 into this equation, and using the property that alternate interior angles are equal, we find this sum to be [tex]\(180^\circ\)[/tex].
Therefore, the proof shows that the sum of the interior angles in a triangle is always [tex]\(180^\circ\)[/tex] as this conclusion is derived using properties that hold for any triangle and parallel lines, not just specific measurements or configurations. This means we have proven it true for:
(A) Every triangle.
