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This problem involves an isosceles triangle with two sides of equal length (7 inches each) and an angle of \( 155^\circ \). We need to find the measures of the remaining angles, specifically angles 1 through 5 as labeled in the diagram.

### Steps to solve:

1. **Identify the isosceles triangle properties:**
   - In an isosceles triangle, the base angles are equal.
   - The sum of the interior angles of any triangle is \( 180^\circ \).

2. **Work with the triangle that contains \( \angle 1 \), \( \angle 2 \), and \( \angle 3 \):**
   - Since angles 1 and 2 are opposite the equal sides, we know \( m\angle 1 = m\angle 2 \).
   - \( m\angle 3 = 155^\circ \) (given).

3. **Calculate angles 1 and 2:**
   The sum of all angles in a triangle is \( 180^\circ \). Thus,
   \[
   m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ
   \]
   Substituting \( m\angle 3 = 155^\circ \):
   \[
   m\angle 1 + m\angle 2 + 155^\circ = 180^\circ
   \]
   \[
   m\angle 1 + m\angle 2 = 25^\circ
   \]
   Since \( m\angle 1 = m\angle 2 \), divide 25° by 2:
   \[
   m\angle 1 = m\angle 2 = 12.5^\circ
   \]

4. **Check other angles:**
   - \( m\angle 4 \) and \( m\angle 5 \) seem to be external angles at the base of the isosceles triangle. Since \( m\angle 1 + m\angle 4 = 180^\circ \) (linear pair), we have:
     \[
     m\angle 4 = 180^\circ - 12.5^\circ = 167.5^\circ
     \]
   - Similarly, for \( m\angle 2 \) and \( m\angle 5 \):
     \[
     m\angle 5 = 180^\circ - 12.5^\circ = 167.5^\circ
     \]

### Final Answer:
- \( m\angle 1 = 12.5^\circ \)
- \( m\angle 2 = 12.5^\circ \)
- \( m\angle 3 = 155^\circ \)
- \( m\angle 4 = 167.5^\circ \)
- \( m\angle 5 = 167.5^\circ \)

Would you like further details on these calculations or have any other questions?

Here are 5 related questions to deepen your understanding:
1. How do we apply the concept of isosceles triangles in different types of problems?
2. What is the exterior angle theorem, and how does it apply to this problem?
3. Can an isosceles triangle have an angle larger than \( 155^\circ \)?
4. How would this problem change if the given angle were different (e.g., \( 120^\circ \))?
5. What are some real-life applications of isosceles triangles?

**Tip:** Always check if angles form a linear pair when dealing with adjacent angles on a straight line to quickly find their measures.

We have seen that isosceles triangles have two sides of equal length. The angles opposite these sides have the same measure. Use the information to the right to help find the measure of angles 1 through 5.

Learn how to calculate the angles of an isosceles triangle when the included angle is 155°. This step-by-step solution involves the properties of isosceles triangles and the linear pair theorem. Suitable for students in Grades 7-9.

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