Question

A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6.
Which statements are always true regarding the diagram? Select three options.

m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠5 + m∠6 =180°
m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 = 180°

Answer 1

To analyze the statements regarding the triangle and its exterior angles, we will use the properties of triangles and exterior angles.

1. **Properties of Triangle Angles:**
- The sum of the interior angles in a triangle is always 180°:
\[
m∠2 + m∠3 + m∠5 = 180°
\]

2. **Exterior Angles:**
- Each exterior angle of the triangle is equal to the sum of the two opposite interior angles. Therefore:
- For angle 1 (exterior angle at angle 2):
\[
m∠1 = m∠3 + m∠5
\]
- For angle 4 (exterior angle at angle 3):
\[
m∠4 = m∠2 + m∠5
\]
- For angle 6 (exterior angle at angle 5):
\[
m∠6 = m∠2 + m∠3
\]

Now, let's evaluate each of the given statements:

1. **m∠5 + m∠3 = m∠4**
- From the exterior angle equation:
\[
m∠4 = m∠2 + m∠5
\]
- Rearranging gives:
\[
m∠4 - m∠5 = m∠2 \implies m∠5 + m∠3 \neq m∠4 \text{ (as angle 1 compares to angle 4)}
\]
- This statement is NOT always true.

2. **m∠3 + m∠4 + m∠5 = 180°**
- This is not valid since \(m∠4\) is an exterior angle based on interior angles.
- Therefore, this statement is NOT true.

3. **m∠5 + m∠6 = 180°**
- Since \(m∠6\) is the exterior angle, it is supplementary to its adjacent interior angle, \(m∠5\):
\[
m∠5 + m∠6 = 180°
\]
- This statement is ALWAYS true.

4. **m∠2 + m∠3 = m∠6**
- From the previous discussion of exterior angles:
\[
m∠6 = m∠2 + m∠3
\]
- This statement is ALWAYS true.

5. **m∠2 + m∠3 + m∠5 = 180°**
- This is the fundamental property of triangles and is true:
\[
m∠2 + m∠3 + m∠5 = 180°
\]
- This statement is ALWAYS true.

Based on the analysis above, the statements that are always true are:

- **m∠5 + m∠6 = 180°**
- **m∠2 + m∠3 = m∠6**
- **m∠2 + m∠3 + m∠5 = 180°**

Thus, the three correct options are:

1. m∠5 + m∠6 = 180°
2. m∠2 + m∠3 = m∠6
3. m∠2 + m∠3 + m∠5 = 180°