Cumulative Standards Review
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Use the fact that the sum of a triangle's angles is 180^(∘).
G
To answer the question, let's consider each option one at a time.
If one of the angles is greater that 90^(∘), then the sum of the two remaining angles is less than 90^(∘). m∠ 1>90^(∘) ⇔ m∠ 2+m∠ 3< 90^(∘) Since the measure of a right angle is 90^(∘), it cannot be one of these angles. Therefore, an obtuse triangle cannot have a right angle.
From the previous section, we know that since one of the triangle's angles is obtuse, the sum of the remaining angles must be less than 90^(∘). An acute angle has a measure less than 90^(∘). There is no other option than for these two remaining angles to be acute. An obtuse triangle can and always will have two acute angles.
According to the definition of an obtuse triangle, it's a triangle with one obtuse angle. An obtuse triangle can and always will have an obtuse angle.
Vertical angles are angles that are opposite to each other when two lines intersect. Let's draw an obtuse triangle and see if it has a pair of vertical angles.
As we can see, ∠ 1 and ∠ 2 are vertical angles. However, angle ∠ 2 is not one of the triangle's angles. Thus, an obtuse triangle (or any other triangle) cannot have two vertical angles.
We come to conclusion that an obtuse triangle can and does have an obtuse angle and two acute angles. The answer is G.