Alex and Heather each draw a triangle with one side of 8cm, one angle of 45° and one angle of 80°.
Alex says that their triangles are congruent.


Alex is correct. The triangles drawn by Alex and Heather are congruent. Here's the reasoning:

### Reasoning:
1. **Given Elements:**
   - One side: 8 cm
   - One angle: 45°
   - Another angle: 80°

2. **Determining the Third Angle:**
   - Since the sum of angles in a triangle is always 180°, the third angle can be calculated as follows:
     \[
     \text{Third Angle} = 180^\circ - 45^\circ - 80^\circ = 55^\circ
     \]
   Therefore, both triangles will have angles of 45°, 80°, and 55°.

3. **Side-Angle-Angle (ASA) Congruence Criterion:**
   - The triangles have a pair of congruent angles (45° and 80°) and the side between these angles (8 cm) is also congruent.
   - By the ASA (Angle-Side-Angle) congruence criterion, the two triangles must be congruent.

### Conclusion:
Since both triangles satisfy the ASA criterion, they are congruent. This means the triangles are identical in shape and size, though they may be oriented differently.

Would you like more details, or do you have any questions?

### Related Questions:
1. What other congruence criteria can be used to determine triangle congruence?
2. How would the problem change if only the two angles were given without a side length?
3. Can you explain why the sum of angles in any triangle is always 180°?
4. What happens if the given angles were different but the side length remained 8 cm?
5. How would the scenario change if the triangles were drawn on different planes?

### Tip:
When two angles and a non-included side of a triangle are known, the triangle's shape and size are uniquely determined if the side is between the two angles.

Explore the concept of triangle congruence using given angles and a side length. Learn how to determine triangle similarity and congruence using the Angle-Side-Angle (ASA) criterion. Suitable for students in Grades 7-9.

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