Question

Select the correct answer. Given: ΔABC Prove: The sum of the interior angle measures of ΔABC is 180°. [edAsset type="image" alt="A line passes through points D, B, and E. A triangle A B C passes through point B on the line. Angles at A and C are 1 and 3. The line forms three angles 4, 2 inside the triangle, and 5 with the triangle." /edAsset] Statement Reason 1. Let points A, B, and C form a triangle. given 2. Let D ⁢ E ⟷ be a line passing through B, parallel to A ⁢ C ¯ , with angles as labeled. defining a parallel line and labeling angles 3. 4. m∠1 = m∠4, and m∠3 = m∠5. Congruent angles have equal measures. 5. m∠4 + m∠2 + m∠5 = 180° angle addition and definition of a straight angle 6. m∠1 + m∠2 + m∠3 = 180° substitution What is the missing step in this proof? A. Statement: ∠4 ≅ ∠5, and ∠1 ≅ ∠3. Reason: Alternate Interior Angles Theorem B. Statement: D ⁢ E ⟷ is parallel to A ⁢ C ¯ . Reason: A ⁢ B ¯ is a transversal cutting D ⁢ E ⟷ and A ⁢ C ¯ . C. Statement: ∠1 ≅ ∠4, and ∠3 ≅ ∠5. Reason: Alternate Interior Angles Theorem D. Statement: ∠1 ≅ ∠4, and ∠3 ≅ ∠5. Reason: ∠1 and ∠4, and ∠3 and ∠5 are pairs of supplementary angles.

Answer 1

The missing proof step states that angles 1 and 4, and angles 3 and 5 are congruent due to the Alternate Interior Angles Theorem because DE is parallel to AC with AB as a transversal.

Step-by-step explanation:

The missing step in the proof that the sum of the interior angles of ∆ABC is 180° is the justification that angles ∠1 and ∠4, and angles ∠3 and ∠5 are congruent. This is true because line DE is drawn parallel to AC, making angles 1 and 4, and angles 3 and 5 pairs of alternate interior angles, which are congruent when the lines are cut by a transversal, in this case, AB. Therefore, the correct answer is: Statement: ∠1 ≅ ∠4, and ∠3 ≅ ∠5. Reason: Alternate Interior Angles Theorem.