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 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.
m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by the definition of a straight angle. Finally, m angle 5 plus m angle 3 plus m angle 4 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 substitution.
The correct answer choice is: m∠1+m∠3+m∠2=180° by the definition of a straight angle. Finally, m∠5+m∠3+m∠4=180° by substitution.
The correct answer choice 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.
To determine the correct answer choice, we need to understand the Triangle Angle Sum Theorem and analyze Sylvie's proof.
Sylvie mentions that ∠1≅∠5 and ∠2≅∠4 because alternate interior angles are congruent when lines are parallel. This means that the given diagram has parallel lines, and angles 1 and 5 are congruent, as well as angles 2 and 4.
Next, Sylvie states that m∠1=m∠5 and m∠2=m∠4 because congruent angles have equal measures. This implies that the measures of angles 1 and 5 are equal, and the measures of angles 2 and 4 are equal.
Now, we need to determine the correct formula to complete Sylvie's proof. The Triangle Angle Sum Theorem states that the sum of the interior angles of a triangle is always 180°. Therefore, to complete the proof, we need to express the sum of the angles in the given diagram as 180°.
The correct answer choice is: m∠1+m∠3+m∠2 = 180° by the definition of a straight angle. Finally, m∠5+m∠3+m∠4 = 180° by substitution.
Explanation: In the given diagram, angle 3 is a common angle for both triangles. Since angles 1 and 2 are congruent to angles 5 and 4, respectively, we can substitute the measures of angles 1, 2, 5, and 4 with their congruent angles. By doing so, we have the equation: m∠1+m∠3+m∠2 = m∠5+m∠3+m∠4, which implies that the sum of the angles in the given diagram is equal to 180°. Therefore, the Triangle Angle Sum Theorem is completed.
Note: The other answer choices either do not use the Triangle Angle Sum Theorem or involve incorrect substitutions.