Question

Prove that the sum of the measures of the exterior angles of the triangle is 360º. Fill in the blanks in step 7 to complete the proof.(1 point)Statement Reason1. Consider a triangle with interior angles ∠2, ∠4, and ∠6 and corresponding exterior angles ∠1, ∠3, and ∠5. 1. given2. m∠1+m∠2=180°m∠5+m∠6=180°m∠3+m∠4=180°2. Definition of Linear Pairs3. (m∠1+m∠2)+(m∠5+m∠6)+(m∠3+m∠4)=180°+180°+180° 3. addition4. (m∠2+m∠4+m∠6)+(m∠1+m∠3+m∠5)=540° 4. rearrange and simplify5. m∠2+m∠4+m∠6=180° 5. Triangle Angle Sum Theorem6. 180°+(m∠1+m∠3+m∠5)=540° 6. substitution7. $$ 7. Subtraction Property of Equality

Answer 1

Let's complete the proof by filling in the blanks for step 7.

1. Consider a triangle with interior angles ∠2, ∠4, and ∠6 and corresponding exterior angles ∠1, ∠3, and ∠5.

   • Reason: Given.

2. \( m∠1 + m∠2 = 180° \), \( m∠5 + m∠6 = 180° \), \( m∠3 + m∠4 = 180° \)

   • Reason: Definition of Linear Pairs.

3. \( (m∠1 + m∠2) + (m∠5 + m∠6) + (m∠3 + m∠4) = 180° + 180° + 180° \)

   • Reason: Addition.

4. \( (m∠2 + m∠4 + m∠6) + (m∠1 + m∠3 + m∠5) = 540° \)

   • Reason: Rearrange and simplify.

5. \( m∠2 + m∠4 + m∠6 = 180° \)

   • Reason: Triangle Angle Sum Theorem.

6. \( 180° + (m∠1 + m∠3 + m∠5) = 540° \)

   • Reason: Substitution.

7. \( m∠1 + m∠3 + m∠5 = 360° \)

   • Reason: Subtraction Property of Equality.

Thus, we have shown that the sum of the measures of the exterior angles of a triangle is \( 360° \).