Question

select true or false for each statement.
a triangle can have more than one right angle.
each vertex of a triangle has two exterior angles, and these angles are always congruent.
the sum of the measures of the interior angles of a triangle is the same as the sum of the exterior angles.
all the exterior angles of a triangle can be obtuse angles.

Explanation:

Step1: Analyze the first statement

A triangle's interior angles sum to \(180^\circ\). A right angle is \(90^\circ\). If there were two right angles, their sum would be \(90^\circ+90^\circ = 180^\circ\), leaving no angle for the third vertex. So a triangle can't have more than one right angle.
<Conclusion: First statement is False>

Step2: Analyze the second statement

At each vertex of a triangle, the two exterior angles are vertical angles (formed by extending different sides), so they are always congruent (equal in measure).
<Conclusion: Second statement is True>

Step3: Analyze the third statement

Sum of interior angles of a triangle is \(180^\circ\). Sum of exterior angles (one at each vertex, considering the linear pair and the fact that for any polygon, sum of exterior angles is \(360^\circ\) for convex polygons, and triangle is convex) is \(360^\circ\). \(180^\circ
eq360^\circ\), so the statement is False.
<Conclusion: Third statement is False>

Step4: Analyze the fourth statement

A triangle can be acute (all interior angles acute, so all exterior angles obtuse, since exterior angle = \(180^\circ -\) interior angle, and if interior is acute (\(<90^\circ\)), exterior is \(>90^\circ\) i.e., obtuse). For example, an equilateral triangle has interior angles \(60^\circ\), so exterior angles \(120^\circ\) (obtuse). So all exterior angles can be obtuse.
<Conclusion: Fourth statement is True>

Answer:

1. A triangle can have more than one right angle: False
2. Each vertex of a triangle has two exterior angles, and these angles are always congruent: True
3. The sum of the measures of the interior angles of a triangle is the same as the sum of the exterior angles: False
4. All the exterior angles of a triangle can be obtuse angles: True