Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
