What is the sum of the measures of the angles of a convex quadrialteral? Will this property hold if the quadrilateral is not convex? (Make a non convex quadrilateral and try!)

Answer

To find the sum of the measures of the angles of a convex quadrilateral and to see if this property holds for a non-convex quadrilateral, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Convex Quadrilateral:**
- A convex quadrilateral is a four-sided figure where all interior angles are less than 180 degrees, and no angles point inward.
- The sum of the interior angles of any quadrilateral can be calculated using the formula:
\[ \text{Sum of angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides. For a quadrilateral, \( n = 4 \).

2. **Calculating the Sum of Angles:**
- Plugging in the value of \( n \):
\[ \text{Sum of angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
- Therefore, the sum of the measures of the angles of a convex quadrilateral is \( 360^\circ \).

3. **Exploring Non-Convex Quadrilaterals:**
- A non-convex quadrilateral (or concave quadrilateral) has at least one interior angle greater than 180 degrees.
- To check if the sum of angles remains the same, we can create a non-convex quadrilateral. For example, consider a quadrilateral with vertices A, B, C, and D where angle A is 270 degrees and the other angles are 90 degrees each.

4. **Calculating Angles in Non-Convex Quadrilateral:**
- Let's assign the angles:
- Angle A = 270 degrees
- Angle B = 90 degrees
- Angle C = 90 degrees
- Angle D = 90 degrees
- Now, we calculate the sum:
\[ \text{Sum of angles} = 270^\circ + 90^\circ + 90^\circ + 90^\circ = 540^\circ \]

5. **Conclusion:**
- The sum of the measures of the angles of a convex quadrilateral is \( 360^\circ \).
- For a non-convex quadrilateral, the sum of the angles can exceed \( 360^\circ \) (as shown in our example, it was \( 540^\circ \)).
- Therefore, the property does not hold for non-convex quadrilaterals.