Question

a) if one angle of a triangle is obtuse, can another also be obtuse? why or why not?
b) if one angle of a triangle is acute, can the other two angles also be acute? why or why not?
c) can a triangle have two right angles? why or why not?
d) if a triangle has one acute angle, is the triangle necessarily acute? why or why not?

a) if one angle of a triangle is obtuse, can another also be obtuse? why or why not? choose the correct answer below.
a. yes, because every obtuse triangle has at least two obtuse angles.
b. no, because the sum of two obtuse angles is more than 180°.
c. yes, because a triangle with two obtuse angles is isosceles.
d. no, because the sum of two obtuse angles is less than 180°.

b) if one angle of a triangle is acute, can the other two angles also be acute? why or why not? choose the correct answer below
a. no, because the other two angles must be obtuse
b. no, because you cant have more than two acute angles in a triangle.
c. yes, because three angles that are less than 90° can sum to 180°
d. yes, because acute angles are supplementary

Explanation:

Part (a)

An obtuse angle is greater than \(90^\circ\) but less than \(180^\circ\). The sum of the interior angles of a triangle is \(180^\circ\). If we have two obtuse angles, say \(\alpha> 90^\circ\) and \(\beta> 90^\circ\), then \(\alpha+\beta> 180^\circ\), and adding the third angle (which is positive) would make the sum of angles greater than \(180^\circ\), which violates the triangle angle - sum property. So, a triangle cannot have two obtuse angles. Option B is correct as it states that the sum of two obtuse angles is more than \(180^\circ\).

An acute angle is less than \(90^\circ\). Let the three angles of a triangle be \(A\), \(B\), and \(C\) with \(A < 90^\circ\), \(B<90^\circ\), and \(C < 90^\circ\). Then \(A + B + C<90^\circ+90^\circ + 90^\circ=270^\circ\), but more importantly, we can find values such that \(A + B + C = 180^\circ\). For example, in an equilateral triangle, all three angles are \(60^\circ\) (which are acute) and their sum is \(180^\circ\). Option A is wrong because the other two angles don't have to be obtuse. Option B is wrong because a triangle can have three acute angles. Option D is wrong because supplementary angles sum to \(180^\circ\), and acute angles are less than \(90^\circ\), so they can't be supplementary. Option C is correct as three angles less than \(90^\circ\) can sum to \(180^\circ\).

Answer:

B. No, because the sum of two obtuse angles is more than \(180^\circ\)

Part (b)