Question

by angles
all angles more than but less than
exactly one angle
but less than
classify each triangle by its angles and sides.
1.
2.
3.
4.
5.
6.
angles of a triangle
interior angles
exterior angles - formed when the sides of the triangle are extended
triangle sum theorem
the sum of the measures of the interior angles of a triangle is

Explanation:

Step1: Recall triangle classification rules

• By angles:
• Acute triangle: All angles are less than \(90^{\circ}\).
• Right triangle: One angle is exactly \(90^{\circ}\).
• Obtuse triangle: One angle is greater than \(90^{\circ}\) and less than \(180^{\circ}\).
• By sides:
• Equilateral triangle: All three sides are equal (all angles \(60^{\circ}\)).
• Isosceles triangle: At least two sides are equal.
• Scalene triangle: All three sides are of different lengths.

Step2: Classify each triangle

Triangle 1:

• Angles: All angles are \(60^{\circ}\) (since \(60 + 60+60 = 180\)).
• Sides: Equal angles imply equal sides (equilateral triangle property).
• Classification: Equilateral (by sides) and acute (by angles).

Triangle 2:

• Angles: One right angle (\(90^{\circ}\)).
• Sides: Two sides are marked equal.
• Classification: Isosceles (by sides) and right - angled (by angles).

Triangle 3:

• Angles: All angles (\(89^{\circ},49^{\circ},42^{\circ}\)) are less than \(90^{\circ}\).
• Sides: No sides are marked equal.
• Classification: Scalene (by sides) and acute (by angles).

Triangle 4:

• Angles: One angle (\(124^{\circ}\)) is greater than \(90^{\circ}\).
• Sides: No sides are marked equal.
• Classification: Scalene (by sides) and obtuse (by angles).

Triangle 5:

• Angles: One angle (\(120^{\circ}\)) is greater than \(90^{\circ}\).
• Sides: Two sides are marked equal ( \(27^{\circ}\) and \(33^{\circ}\) angles, the sides opposite to non - equal angles are equal as per the rule in isosceles triangles where angles opposite to equal sides are equal).
• Classification: Isosceles (by sides) and obtuse (by angles).

Triangle 6:

• Angles: One right angle (\(90^{\circ}\)).
• Sides: No sides are marked equal.
• Classification: Scalene (by sides) and right - angled (by angles).

Step3: Triangle Sum Theorem

The sum of the measures of the interior angles of a triangle is \(180^{\circ}\). This is a fundamental property of triangles, which can be proved using various methods such as drawing a parallel line to one side of the triangle and using the properties of alternate and corresponding angles.

Answer:

1. Equilateral (by sides), Acute (by angles)
2. Isosceles (by sides), Right - angled (by angles)
3. Scalene (by sides), Acute (by angles)
4. Scalene (by sides), Obtuse (by angles)
5. Isosceles (by sides), Obtuse (by angles)
6. Scalene (by sides), Right - angled (by angles)

Sum of interior angles: \(180^{\circ}\)