Question

in exercises 3–6, classify the triangle by its sides and by measuring its angles. (see example 1.)
3.
triangle xyz with right angle at y, xy and yz marked congruent
4.
triangle lmn with all sides marked congruent
5.
triangle hjk, scalene?
6.
triangle abc, scalene?

Explanation:

To solve this, we analyze each triangle:

Problem 3 (Triangle \( XYZ \)):

• By Sides: Two sides (\( XY \) and \( YZ \)) are marked congruent (equal length), so it is an isosceles triangle.
• By Angles: \( \angle Y \) is a right angle (\( 90^\circ \)), so it is a right triangle.

Thus, triangle \( XYZ \) is an isosceles right triangle.

Problem 4 (Triangle \( LMN \)):

• By Sides: All three sides are marked congruent (equal length), so it is an equilateral triangle.
• By Angles: In an equilateral triangle, all angles are \( 60^\circ \) (acute), so it is also an acute triangle.

Problem 5 (Triangle \( HJK \)):

• By Sides: Two sides appear congruent (visual inspection), so it is an isosceles triangle.
• By Angles: One angle is obtuse (greater than \( 90^\circ \)), so it is an obtuse triangle.

Problem 6 (Triangle \( ABC \)):

• By Sides: All three sides are of different lengths (scalene), so it is a scalene triangle.
• By Angles: All angles are acute (less than \( 90^\circ \)), so it is an acute triangle.

Final Answers:

1. Isosceles Right Triangle
2. Equilateral (Acute) Triangle
3. Isosceles Obtuse Triangle
4. Scalene Acute Triangle

Answer:

To solve this, we analyze each triangle:

Problem 3 (Triangle \( XYZ \)):

• By Sides: Two sides (\( XY \) and \( YZ \)) are marked congruent (equal length), so it is an isosceles triangle.
• By Angles: \( \angle Y \) is a right angle (\( 90^\circ \)), so it is a right triangle.

Thus, triangle \( XYZ \) is an isosceles right triangle.

Problem 4 (Triangle \( LMN \)):

• By Sides: All three sides are marked congruent (equal length), so it is an equilateral triangle.
• By Angles: In an equilateral triangle, all angles are \( 60^\circ \) (acute), so it is also an acute triangle.

Problem 5 (Triangle \( HJK \)):

• By Sides: Two sides appear congruent (visual inspection), so it is an isosceles triangle.
• By Angles: One angle is obtuse (greater than \( 90^\circ \)), so it is an obtuse triangle.

Problem 6 (Triangle \( ABC \)):

• By Sides: All three sides are of different lengths (scalene), so it is a scalene triangle.
• By Angles: All angles are acute (less than \( 90^\circ \)), so it is an acute triangle.

Final Answers:

1. Isosceles Right Triangle
2. Equilateral (Acute) Triangle
3. Isosceles Obtuse Triangle
4. Scalene Acute Triangle