Question

for each triangle, check all that apply.
triangle a
triangle b
triangle c
triangle d
scalene
isosceles
equilateral
scalene
isosceles
equilateral
scalene
isosceles
equilateral
scalene
isosceles
equilateral

Explanation:

To solve this, we analyze each triangle based on side lengths (for scalene, isosceles, equilateral) and angle measures:

Triangle A

• Angles: \( 45^\circ, 75^\circ, 60^\circ \) (all angles are different, so all sides are different).
• Scalene: A triangle with all sides of different lengths (implied by all angles different).
• Isosceles: Requires at least two equal sides (no, since all angles are different).
• Equilateral: Requires all angles \( 60^\circ \) (no, angles are \( 45^\circ, 75^\circ, 60^\circ \)).

Triangle B

• Side lengths: \( 5, 7, 11 \) (all sides are different).
• Scalene: A triangle with all sides of different lengths.
• Isosceles: Requires at least two equal sides (no, all sides are different).
• Equilateral: Requires all sides equal (no, sides are \( 5, 7, 11 \)).

Triangle C

• Angles: \( 40^\circ, 40^\circ, 100^\circ \) (two angles are equal, so two sides are equal).
• Scalene: Requires all sides different (no, two sides are equal).
• Isosceles: A triangle with at least two equal sides (implied by two equal angles).
• Equilateral: Requires all angles \( 60^\circ \) (no, angles are \( 40^\circ, 40^\circ, 100^\circ \)).

Triangle D

• Side lengths: \( 7, 7, 7 \) (all sides equal).
• Scalene: Requires all sides different (no, all sides are equal).
• Isosceles: A triangle with at least two equal sides (yes, all three are equal, so it is also isosceles, but equilateral is a special case of isosceles).
• Equilateral: A triangle with all sides equal (yes, all sides are \( 7 \)).

Final Checks:

• Triangle A: Check \(\boldsymbol{\text{Scalene}}\).
• Triangle B: Check \(\boldsymbol{\text{Scalene}}\).
• Triangle C: Check \(\boldsymbol{\text{Isosceles}}\).
• Triangle D: Already checked \(\boldsymbol{\text{Equilateral}}\) (and also \(\text{Isosceles}\), but the primary check here is Equilateral, and Isosceles is also true, but the problem asks to “check all that apply”).

Completed Table:

Triangle	Scalene	Isosceles	Equilateral
B	✔️	❌	❌
C	❌	✔️	❌
D	❌	✔️	✔️

(Note: For Triangle D, “Isosceles” is also true because equilateral triangles are a subset of isosceles triangles, but the problem likely expects “Equilateral” as the primary check here, and “Isosceles” is an additional correct check.)

Answer:

To solve this, we analyze each triangle based on side lengths (for scalene, isosceles, equilateral) and angle measures:

Triangle A

• Angles: \( 45^\circ, 75^\circ, 60^\circ \) (all angles are different, so all sides are different).
• Scalene: A triangle with all sides of different lengths (implied by all angles different).
• Isosceles: Requires at least two equal sides (no, since all angles are different).
• Equilateral: Requires all angles \( 60^\circ \) (no, angles are \( 45^\circ, 75^\circ, 60^\circ \)).

Triangle B

• Side lengths: \( 5, 7, 11 \) (all sides are different).
• Scalene: A triangle with all sides of different lengths.
• Isosceles: Requires at least two equal sides (no, all sides are different).
• Equilateral: Requires all sides equal (no, sides are \( 5, 7, 11 \)).

Triangle C

• Angles: \( 40^\circ, 40^\circ, 100^\circ \) (two angles are equal, so two sides are equal).
• Scalene: Requires all sides different (no, two sides are equal).
• Isosceles: A triangle with at least two equal sides (implied by two equal angles).
• Equilateral: Requires all angles \( 60^\circ \) (no, angles are \( 40^\circ, 40^\circ, 100^\circ \)).

Triangle D

• Side lengths: \( 7, 7, 7 \) (all sides equal).
• Scalene: Requires all sides different (no, all sides are equal).
• Isosceles: A triangle with at least two equal sides (yes, all three are equal, so it is also isosceles, but equilateral is a special case of isosceles).
• Equilateral: A triangle with all sides equal (yes, all sides are \( 7 \)).

Final Checks:

• Triangle A: Check \(\boldsymbol{\text{Scalene}}\).
• Triangle B: Check \(\boldsymbol{\text{Scalene}}\).
• Triangle C: Check \(\boldsymbol{\text{Isosceles}}\).
• Triangle D: Already checked \(\boldsymbol{\text{Equilateral}}\) (and also \(\text{Isosceles}\), but the primary check here is Equilateral, and Isosceles is also true, but the problem asks to “check all that apply”).

Completed Table:

Triangle	Scalene	Isosceles	Equilateral
B	✔️	❌	❌
C	❌	✔️	❌
D	❌	✔️	✔️

(Note: For Triangle D, “Isosceles” is also true because equilateral triangles are a subset of isosceles triangles, but the problem likely expects “Equilateral” as the primary check here, and “Isosceles” is an additional correct check.)