Question
Use the image to answer the question.
An illustration shows a triangle with angles marked as 1, 2 and 3, clockwise beginning at the top angle. A line is drawn outside of the triangle, passing through the vertex with angle 3.
Sylvie has started a proof of the Triangle Angle Sum Theorem. Which answer choice correctly completes her proof?
Sylvie's Proof: Given the diagram shown, ∠1≅∠5; ∠2≅∠4 because alternate interior angles are congruent when lines are parallel. Then, I know that m∠1=m∠5; m∠2=m∠4 because congruent angles have equal measures.
(1 point)
Responses
m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution.
m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by substitution.
m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by the Triangle Angle Sum Theorem.
m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by the Triangle Angle Sum Theorem.
m∠2+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution.
m angle 2 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by substitution.
m∠1+m∠3+m∠2=180° by the definition of a straight angle. Finally, m∠5+m∠3+m∠4=180° by substitution.
An illustration shows a triangle with angles marked as 1, 2 and 3, clockwise beginning at the top angle. A line is drawn outside of the triangle, passing through the vertex with angle 3.
Sylvie has started a proof of the Triangle Angle Sum Theorem. Which answer choice correctly completes her proof?
Sylvie's Proof: Given the diagram shown, ∠1≅∠5; ∠2≅∠4 because alternate interior angles are congruent when lines are parallel. Then, I know that m∠1=m∠5; m∠2=m∠4 because congruent angles have equal measures.
(1 point)
Responses
m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution.
m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by substitution.
m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by the Triangle Angle Sum Theorem.
m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by the Triangle Angle Sum Theorem.
m∠2+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution.
m angle 2 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by substitution.
m∠1+m∠3+m∠2=180° by the definition of a straight angle. Finally, m∠5+m∠3+m∠4=180° by substitution.
Answers
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Answered by
GPT-4o mini
To complete Sylvie's proof of the Triangle Angle Sum Theorem effectively, we need to ensure it flows logically from her established points.
Sylvie has established that:
1. ∠1≅∠5 and ∠2≅∠4 due to the alternate interior angles being congruent when the lines are parallel.
2. This leads to m∠1 = m∠5 and m∠2 = m∠4.
From Sylvie's proof, she can conclude that the angles formed outside the triangle along with the triangle's interior angles should sum up to 180° since they are on a straight line.
The correct statement to complete her proof would be:
**m∠5 + m∠3 + m∠4 = 180° by the definition of a straight angle. Finally, m∠1 + m∠3 + m∠2 = 180° by substitution.**
So the correct answer is:
**m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution.**
Sylvie has established that:
1. ∠1≅∠5 and ∠2≅∠4 due to the alternate interior angles being congruent when the lines are parallel.
2. This leads to m∠1 = m∠5 and m∠2 = m∠4.
From Sylvie's proof, she can conclude that the angles formed outside the triangle along with the triangle's interior angles should sum up to 180° since they are on a straight line.
The correct statement to complete her proof would be:
**m∠5 + m∠3 + m∠4 = 180° by the definition of a straight angle. Finally, m∠1 + m∠3 + m∠2 = 180° by substitution.**
So the correct answer is:
**m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution.**
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