Question

matching list
question 3
listen
consider the four triangles below. the triangles are not necessarily drawn to scale.
geometric figures
a
b
c
d
85°
60°
55°
60°
40°
100°
60°
what is the most accurate description of each of the triangles?

• match the description to the correct triangle based on the letter next to the triangle.

Explanation:

To solve this, we analyze each triangle's angles (and sides, if applicable) to classify them (e.g., right, isosceles, equilateral, acute, obtuse) using the triangle angle - sum property ($180^\circ$) and side - angle relationships.

Triangle A

Step 1: Identify the right angle

Triangle A has a right angle ($90^\circ$). Let the other two angles be $x$ and $y$. By the triangle angle - sum property, $x + y+90^\circ= 180^\circ$, so $x + y = 90^\circ$. Also, if we assume it's a right - isosceles triangle (from the diagram's appearance, the legs seem equal), the other two angles are equal. So $x=y=\frac{90^\circ}{2} = 45^\circ$. So Triangle A is a right (and isosceles) triangle.

Triangle B

Step 1: Find the third angle

We know one angle is $100^\circ$. Let the other two angles be $m$ and $n$. Using the angle - sum property: $m + n+100^\circ=180^\circ$, so $m + n = 80^\circ$. Since $100^\circ>90^\circ$, Triangle B is an obtuse triangle. Also, the other two angles are not equal (from the diagram's shape, the sides opposite them seem unequal), so it's a scalene obtuse triangle.

Triangle C

Step 1: Verify angle - sum

The angles are $55^\circ$, $85^\circ$, and $40^\circ$. Let's check the sum: $55^\circ+85^\circ + 40^\circ=180^\circ$ (since $55 + 85=140$, and $140 + 40 = 180$). All angles are less than $90^\circ$, so Triangle C is an acute triangle. Also, no two angles are equal (so sides opposite are unequal), so it's a scalene acute triangle.

Triangle D

Step 1: Analyze angles

All three angles are $60^\circ$. Since all angles are equal, by the converse of the isosceles triangle theorem, all sides are equal. So Triangle D is an equilateral (and also acute, and isosceles) triangle.

If we assume the matching options are related to triangle types (e.g., right triangle, obtuse triangle, acute triangle, equilateral triangle):

• Triangle A: Right (Isosceles) Triangle
• Triangle B: Obtuse (Scalene) Triangle
• Triangle C: Acute (Scalene) Triangle
• Triangle D: Equilateral (Acute, Isosceles) Triangle

If we were to match with typical descriptions:

• A: Right Triangle (because of the $90^\circ$ angle)
• B: Obtuse Triangle (because of the $100^\circ$ angle)
• C: Acute Triangle (all angles $<90^\circ$)
• D: Equilateral Triangle (all angles $ = 60^\circ$)

For example, if the matching list has options like "Right Triangle", "Obtuse Triangle", "Acute Triangle", "Equilateral Triangle":

• A matches "Right Triangle"
• B matches "Obtuse Triangle"
• C matches "Acute Triangle"
• D matches "Equilateral Triangle"

Answer:

To solve this, we analyze each triangle's angles (and sides, if applicable) to classify them (e.g., right, isosceles, equilateral, acute, obtuse) using the triangle angle - sum property ($180^\circ$) and side - angle relationships.

Triangle A

Step 1: Identify the right angle

Triangle A has a right angle ($90^\circ$). Let the other two angles be $x$ and $y$. By the triangle angle - sum property, $x + y+90^\circ= 180^\circ$, so $x + y = 90^\circ$. Also, if we assume it's a right - isosceles triangle (from the diagram's appearance, the legs seem equal), the other two angles are equal. So $x=y=\frac{90^\circ}{2} = 45^\circ$. So Triangle A is a right (and isosceles) triangle.

Triangle B

Step 1: Find the third angle

We know one angle is $100^\circ$. Let the other two angles be $m$ and $n$. Using the angle - sum property: $m + n+100^\circ=180^\circ$, so $m + n = 80^\circ$. Since $100^\circ>90^\circ$, Triangle B is an obtuse triangle. Also, the other two angles are not equal (from the diagram's shape, the sides opposite them seem unequal), so it's a scalene obtuse triangle.

Triangle C

Step 1: Verify angle - sum

The angles are $55^\circ$, $85^\circ$, and $40^\circ$. Let's check the sum: $55^\circ+85^\circ + 40^\circ=180^\circ$ (since $55 + 85=140$, and $140 + 40 = 180$). All angles are less than $90^\circ$, so Triangle C is an acute triangle. Also, no two angles are equal (so sides opposite are unequal), so it's a scalene acute triangle.

Triangle D

Step 1: Analyze angles

All three angles are $60^\circ$. Since all angles are equal, by the converse of the isosceles triangle theorem, all sides are equal. So Triangle D is an equilateral (and also acute, and isosceles) triangle.

If we assume the matching options are related to triangle types (e.g., right triangle, obtuse triangle, acute triangle, equilateral triangle):

• Triangle A: Right (Isosceles) Triangle
• Triangle B: Obtuse (Scalene) Triangle
• Triangle C: Acute (Scalene) Triangle
• Triangle D: Equilateral (Acute, Isosceles) Triangle

If we were to match with typical descriptions:

• A: Right Triangle (because of the $90^\circ$ angle)
• B: Obtuse Triangle (because of the $100^\circ$ angle)
• C: Acute Triangle (all angles $<90^\circ$)
• D: Equilateral Triangle (all angles $ = 60^\circ$)

For example, if the matching list has options like "Right Triangle", "Obtuse Triangle", "Acute Triangle", "Equilateral Triangle":

• A matches "Right Triangle"
• B matches "Obtuse Triangle"
• C matches "Acute Triangle"
• D matches "Equilateral Triangle"