Questions: Prove that the measures of the exterior angles of a triangle have a sum of 360°. Given: angle 1, angle 2, and angle 3 are exterior angles. Prove: m angle 1 + m angle 2 + m angle 3 = 360 It is given that angle 1, angle 2, and angle 3 are exterior angles.

Prove that the measures of the exterior angles of a triangle have a sum of 360°.
Given: angle 1, angle 2, and angle 3 are exterior angles.
Prove: m angle 1 + m angle 2 + m angle 3 = 360

It is given that angle 1, angle 2, and angle 3 are exterior angles.
Transcript text: Prove that the measures of the exterior angles of a triangle have a sum of $360^{\circ}$. Given: $\angle 1, \angle 2$, and $\angle 3$ are exterior angles. Prove: $m \angle 1+m \angle 2+m \angle 3=360$ It is given that $\angle 1, \angle 2$, and $\angle 3$ are exterior angles.
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Solution

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Solution Steps

Step 1: Identify the Exterior Angles

Given that ∠1, ∠2, and ∠3 are exterior angles of the triangle.

Step 2: Apply the Triangle Exterior Angle Theorem

By the Triangle Exterior Angle Theorem, each exterior angle is equal to the sum of the two non-adjacent interior angles. Therefore:

  • m∠1 = 180° - m∠A
  • m∠2 = 180° - m∠B
  • m∠3 = 180° - m∠C
Step 3: Sum of Exterior Angles

The sum of the measures of the exterior angles of a triangle is: m∠1 + m∠2 + m∠3 = (180° - m∠A) + (180° - m∠B) + (180° - m∠C)

Step 4: Simplify the Expression

Combine the terms: m∠1 + m∠2 + m∠3 = 180° + 180° + 180° - (m∠A + m∠B + m∠C) m∠1 + m∠2 + m∠3 = 540° - (m∠A + m∠B + m∠C)

Step 5: Use the Sum of Interior Angles of a Triangle

The sum of the interior angles of a triangle is always 180°: m∠A + m∠B + m∠C = 180°

Step 6: Substitute and Finalize

Substitute the sum of the interior angles into the equation: m∠1 + m∠2 + m∠3 = 540° - 180° m∠1 + m∠2 + m∠3 = 360°

Final Answer

The sum of the measures of the exterior angles of a triangle is 360°.

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