Answer :
A convex pentagon can have at most three of its interior angles measuring 90 degrees.
To answer this question, we need to use the fact that the sum of the interior angles of a polygon can be determined using the formula:
Sum of interior angles = (n - 2) × 180°, where n is the number of sides of the polygon.
For a pentagon (n = 5), the sum of the interior angles is:
(5 - 2) × 180° = 3 × 180° = 540°
Since each right angle is 90°, we can now determine the possible number of 90° angles. Let’s assume k of the angles are 90°:
k × 90° + remaining angles = 540°
The remaining angles sum to:
540° - k × 90°
These remaining angles must also be positive, and since we're dealing with a pentagon (five angles in total), k cannot exceed 4 (though let's verify what happens if we try more).
- If k = 4:
4 × 90° + remaining angle = 540°
360° + remaining angle = 540°
remaining angle = 180°
However, an angle of 180° would make the polygon not convex, since a convex polygon's interior angles must all be less than 180°.
- If k = 3:
3 × 90° + 2 remaining angles = 540°
270° + 2 remaining angles = 540°
2 remaining angles = 270°
Each remaining angle = 270° ÷ 2 = 135°
Therefore, a convex pentagon can have at most three 90° angles, with the other two angles being 135° each.
Answer:
3
Step-by-step explanation:
To determine the largest possible number of interior angles of a convex pentagon that can measure 90°, we first need to consider the sum of the interior angles of a pentagon.
The formula for the sum of the interior angles of a polygon is:
[tex]\boxed{\begin{array}{c}\underline{\textsf{Sum of the interior angles of a polygon}}\\\\S=(n-2) \times 180^{\circ}\\\\\textsf{where $n$ is the number of sides} \end{array}}[/tex]
A pentagon has 5 sides, so substitute n = 5 into the formula:
[tex]S=(5-2)\times 180^{\circ}\\\\S = 3\times 180^{\circ}\\\\S = 540^{\circ}[/tex]
Therefore, the sum of the interior angles of a pentagon is 540°.
Let x be the number of 90° angles in the pentagon.
The sum of the x right angles is:
[tex]90^{\circ}\times x=90^{\circ}x[/tex]
So, the sum of the remaining 5 - x angles is:
[tex]540^{\circ}-90^{\circ}x[/tex]
A convex polygon is a polygon in which all interior angles are less than 180°.
Therefore, for the pentagon to be convex, each of the remaining angles must be less than 180°. This gives the inequality:
[tex]540^{\circ}-90^{\circ}x < 180^{\circ}(5-x)[/tex]
Solve for x:
[tex]540-90x < 180(5-x)\\\\\\540-90x < 900-180x\\\\\\-90x+180x < 900-540\\\\\\90x < 360\\\\\\x < \dfrac{360}{90}\\\\\\x < 4[/tex]
As the number of 90° angles is less than 4, the largest possible number of interior angles of a convex pentagon that can measure 90° is:
[tex]\Huge\boxed{\boxed{3}}[/tex]
