A pentagon has two right angles, a 100° angle and a 120° angle. What is the measure of its fifth angle
The Correct Answer and Explanation is:
A pentagon has 5 sides. The formula to find the sum of the interior angles of an n-sided polygon is:Sum of interior angles=(n−2)×180∘\text{Sum of interior angles} = (n – 2) \times 180^\circSum of interior angles=(n−2)×180∘
Substituting n=5n = 5n=5:(5−2)×180∘=3×180∘=540∘(5 – 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ(5−2)×180∘=3×180∘=540∘
So, the total sum of the interior angles in a pentagon is 540 degrees.
Now, we are told that four of the five angles are:
- Two right angles: 90∘+90∘=180∘90^\circ + 90^\circ = 180^\circ90∘+90∘=180∘
- One angle of 100∘100^\circ100∘
- One angle of 120∘120^\circ120∘
Let us now add these known angles:90∘+90∘+100∘+120∘=400∘90^\circ + 90^\circ + 100^\circ + 120^\circ = 400^\circ90∘+90∘+100∘+120∘=400∘
Since the total must be 540°, we subtract the sum of the known angles from 540° to find the measure of the fifth angle:540∘−400∘=140∘540^\circ – 400^\circ = 140^\circ540∘−400∘=140∘
Final Answer:
The measure of the fifth angle is 140 degrees.
Explanation
In geometry, the measure of the interior angles of a polygon can be determined using a standard formula. For any polygon with n sides, the sum of the interior angles is given by:(n−2)×180∘(n – 2) \times 180^\circ(n−2)×180∘
This formula comes from the fact that any polygon can be divided into (n−2)(n – 2)(n−2) triangles, and each triangle has an angle sum of 180°. For a pentagon, which has five sides, the sum of all interior angles is:(5−2)×180∘=540∘(5 – 2) \times 180^\circ = 540^\circ(5−2)×180∘=540∘
We are given four of the five interior angles: two of them are right angles, meaning they measure 90° each, and the other two are 100° and 120°, respectively. By adding these known angles, we get:90∘+90∘+100∘+120∘=400∘90^\circ + 90^\circ + 100^\circ + 120^\circ = 400^\circ90∘+90∘+100∘+120∘=400∘
Since the entire interior angle sum must be 540°, the measure of the missing, or fifth, angle must be the difference between 540° and the total of the known angles:540∘−400∘=140∘540^\circ – 400^\circ = 140^\circ540∘−400∘=140∘
Therefore, the fifth angle in the pentagon must measure 140 degrees. This ensures that the total of all five interior angles adds up correctly to 540°, satisfying the geometric property of a pentagon.
