Question

ana tried to prove that an exterior triangle angle measure equals the sum of the measures of the two interior angles not adjacent to it.

triangle and linear pair diagram

statement	reason
2 ( mangle3 + mangle4 = 180^circ )	the angle measures of a linear pair sum to ( 180^circ ).
3 ( mangle1 + mangle2 + mangle3 = mangle3 + mangle4 )	substitution
4 ( mangle1 + mangle2 = mangle4 )	subtract ( mangle3 ).

what was the first mistake in ana’s proof?

choose 1 answer:

a ( mangle1 + mangle2 + mangle3 = 180^circ ) was not given.

b ( angle3 ) and ( angle4 ) aren’t a linear pair.

c the substitution isn’t correct.

d we can’t subtract an angle measure from both sides of an equation.

Explanation:

To determine the first mistake in Ana's proof, we analyze each option:

• Option A: The sum of the interior angles of a triangle is \(180^\circ\), which is a theorem (Triangle Angle Sum Theorem), not a "given" in the context of proving the exterior angle theorem. So stating \(m\angle1 + m\angle2 + m\angle3 = 180^\circ\) as "Given" is incorrect because it should be justified by the triangle angle sum theorem, not taken as given.
• Option B: \(\angle3\) and \(\angle4\) are adjacent and form a straight line, so they are a linear pair. This reason is correct.
• Option C: Substitution is valid here since both sides equal \(180^\circ\), so replacing one with the other is correct.
• Option D: Subtracting the same angle measure from both sides of an equation is a valid algebraic operation (Subtraction Property of Equality), so this is not a mistake.

Answer:

A. \( m\angle1 + m\angle2 + m\angle3 = 180^\circ \) was not given.