Question

1. given \\( \triangle jkl \\) with \\( m\angle j = 63^\circ \\) and \\( m\angle l = 54^\circ \\), is the triangle an acute, isosceles, obtuse, or right triangle?

type your answer in the box.
the triangle jkl is a triangle triangle

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^\circ\). Let \(m\angle K\) be the third angle. So, \(m\angle K = 180^\circ - m\angle J - m\angle L\). Substituting \(m\angle J = 63^\circ\) and \(m\angle L = 54^\circ\), we get \(m\angle K = 180 - 63 - 54 = 63^\circ\).

Step2: Classify by angles (acute, obtuse, right)

All angles (\(63^\circ\), \(54^\circ\), \(63^\circ\)) are less than \(90^\circ\), so it's acute.

Step3: Classify by sides (isosceles, etc.)

Two angles (\(\angle J\) and \(\angle K\)) are equal, so the sides opposite them are equal (by the Isosceles Triangle Theorem). So, it's isosceles. Combining, it's an acute isosceles triangle. But the question asks for the type (acute, isosceles, obtuse, right) – since all angles are acute and two sides equal, but the main angle - based classification for "acute" (since all angles <90) and side - based "isosceles" (two equal angles imply two equal sides). But the problem might be asking for the angle - type (acute) or the side - type (isosceles). Wait, the angles: all are acute, so the triangle is acute. Also, since two angles are equal, it's isosceles. But let's check the steps again.

Wait, the problem says "is the triangle an acute, isosceles, obtuse, or right triangle?" So we need to see which categories it falls into. Let's re - evaluate:

1. Acute: All angles < \(90^\circ\) – yes, \(63\), \(54\), \(63\) are all < \(90\).
2. Isosceles: Two angles equal (so two sides equal) – yes, \(\angle J\) and \(\angle K\) are \(63^\circ\).
3. Obtuse: Has an angle > \(90^\circ\) – no.
4. Right: Has a \(90^\circ\) angle – no.

But maybe the question is asking for the primary angle - based classification or the side - based. But since it's a triangle with all acute angles and isosceles, but let's see the calculation again.

Wait, the sum of angles: \(63 + 54+63 = 180\), correct. All angles are acute, and two angles are equal (so isosceles). But the problem might be expecting "acute" or "isosceles". Wait, maybe the question is asking for the type (acute, isosceles, etc.). Let's check the angles:

Since all angles are less than \(90^\circ\), it's an acute triangle. Also, since two angles are equal, it's isosceles. But maybe the answer is "acute isosceles", but let's see the problem statement again. The problem says "is the triangle an acute, isosceles, obtuse, or right triangle?" So we need to determine which of these it is. Since it's both acute and isosceles, but maybe the question is asking for the angle - type (acute) or the side - type (isosceles). But let's go back to the steps.

First, find the third angle: \(180-(63 + 54)=63\) degrees. So angles are \(63\), \(54\), \(63\). All angles are acute (less than \(90\)), so it's an acute triangle. Also, two angles are equal, so it's isosceles. But maybe the answer is "acute" (since the question lists acute, isosceles, etc. as options). Wait, the problem might be asking for the angle - based classification (acute) or the side - based (isosceles). But let's see the original problem again.

The problem says "is the triangle an acute, isosceles, obtuse, or right triangle?" So we have to check each:

• Acute: Yes, all angles < \(90\).
• Isosceles: Yes, two angles equal.
• Obtuse: No.
• Right: No.

But maybe the question is asking for the most appropriate single classification, or maybe it's a multiple - choice - like question where we have to pick. But based on the calculation, the triangle is acute (because all angles are acute) and isosceles (because two angles are equal). But let's see the steps again.

Answer:

acute (and isosceles, but if we have to choose one of the given types: acute, since all angles are acute)