Question

complete the proof that an exterior triangle angle measure equals the sum of the measures of the two interior angles not adjacent to it.

triangle diagram with points a, b, c, d; angle at a is 60°, angle at c is 40°, angles at b labeled 1 and 2

statement	reason
2 ( mangle 1 = ) dropdown	substitute and solve for ( mangle 1 ).
3 ( mangle 1 + mangle 2 = 180^circ )	the angle measures of a linear pair sum to ( 180^circ ).
4 ( mangle 2 = ) dropdown	substitute and solve for ( mangle 2 ).
5 ( mangle a + mangle c = 100^circ )	add given angle measures.
6 ( mangle 2 = mangle a + mangle c )	substitution

did we prove the conclusion true for every triangle or only for this specific triangle?

choose 1 answer:

a every triangle

b only for this specific triangle

Explanation:

Step1: Solve for \( m\angle1 \)

We know \( m\angle1 + m\angle A + m\angle C = 180^\circ \), \( m\angle A = 60^\circ \), \( m\angle C = 40^\circ \). Substitute these values: \( m\angle1 + 60^\circ + 40^\circ = 180^\circ \). Simplify: \( m\angle1 + 100^\circ = 180^\circ \), so \( m\angle1 = 180^\circ - 100^\circ = 80^\circ \).

Step2: Solve for \( m\angle2 \)

We know \( m\angle1 + m\angle2 = 180^\circ \) and \( m\angle1 = 80^\circ \). Substitute: \( 80^\circ + m\angle2 = 180^\circ \), so \( m\angle2 = 180^\circ - 80^\circ = 100^\circ \).

Step3: Determine the proof's scope

The steps use the general triangle angle - sum property (\( m\angle1 + m\angle A + m\angle C = 180^\circ \)) and the linear pair property (\( m\angle1 + m\angle2 = 180^\circ \)), which hold for any triangle. Even though specific angle measures (\( 60^\circ \) and \( 40^\circ \)) were used in substitution, the reasoning is based on general properties, so the conclusion holds for every triangle.

Answer:

• For \( m\angle1 \): \( 80^\circ \)
• For \( m\angle2 \): \( 100^\circ \)
• For the proof's scope: A. Every triangle