Question

determine whether each statement about angle relationships of a triangle is true.
select true or false for each statement.
a right triangle always has obtuse exterior angles at two vertices.
the sum of the measures of exterior angles minus the sum of the measures of interior angles is 180°.
the sum of an exterior angle and the two nonadjacent interior angles is 180°.
an obtuse triangle always has only one vertex with an acute exterior angle.

Explanation:

1. Analyze the statement "A right triangle always has obtuse exterior angles at two vertices"

• A right - triangle has one right angle (\(90^{\circ}\)) and two acute angles (less than \(90^{\circ}\)).
• The exterior angle of an angle \(x\) is \(180 - x\).
• For the two acute interior angles \(x\) (where \(0
• For the right - angle interior angle (\(x = 90^{\circ}\)), the exterior angle \(y=180 - 90=90^{\circ}\) (a right - angle, not obtuse). But the statement says "at two vertices". Since the two non - right interior angles (acute) will have obtuse exterior angles. So this statement is True.

2. Analyze the statement "The sum of the measures of exterior angles minus the sum of the measures of interior angles is \(180^{\circ}\)"

• The sum of the interior angles of a triangle is \(S_{i}=(3 - 2)\times180^{\circ}=180^{\circ}\).
• The sum of the exterior angles of any polygon is \(S_{e}=360^{\circ}\).
• Then \(S_{e}-S_{i}=360^{\circ}-180^{\circ}=180^{\circ}\). So this statement is True.

3. Analyze the statement "The sum of an exterior angle and the two non - adjacent interior angles is \(180^{\circ}\)"

• Let the exterior angle be \(E\) and the adjacent interior angle be \(I\). We know that \(E + I=180^{\circ}\) (linear pair). Also, by the angle - sum property of a triangle \(I+A + B=180^{\circ}\) (where \(A\) and \(B\) are the non - adjacent interior angles). Then \(E=A + B\). And \(E+A + B=(A + B)+A + B

eq180^{\circ}\) (in general). The correct relationship is that an exterior angle is equal to the sum of the two non - adjacent interior angles. So this statement is False.

4. Analyze the statement "An obtuse triangle always has only one vertex with an acute exterior angle"

• In an obtuse triangle, there is one obtuse interior angle (greater than \(90^{\circ}\)) and two acute interior angles (less than \(90^{\circ}\)).
• The exterior angle of an obtuse interior angle \(O\) (where \(O>90^{\circ}\)) is \(180 - O<90^{\circ}\) (acute). The exterior angles of the two acute interior angles \(A_{1}\) and \(A_{2}\) (where \(A_{1}<90^{\circ}\) and \(A_{2}<90^{\circ}\)) are \(180 - A_{1}>90^{\circ}\) and \(180 - A_{2}>90^{\circ}\) (obtuse). So an obtuse triangle always has only one vertex with an acute exterior angle. This statement is True.

Answer:

1. True
2. True
3. False
4. True