Find the value of x in the triangle shown below. X = 42° 106
The Correct Answer and Explanation is:
To solve for the missing angle xxx in a triangle, we use a fundamental property of triangles:
The sum of the interior angles of a triangle is always 180°.
We are given two angles of the triangle:
- One angle is xxx degrees (unknown)
- One angle is 42∘42^\circ42∘
- One angle is 106∘106^\circ106∘
Using the rule for the sum of the interior angles:x+42∘+106∘=180∘x + 42^\circ + 106^\circ = 180^\circx+42∘+106∘=180∘
Now, add the known angles:42∘+106∘=148∘42^\circ + 106^\circ = 148^\circ42∘+106∘=148∘
Subtract this sum from 180 degrees to find the missing angle:x=180∘−148∘=32∘x = 180^\circ – 148^\circ = 32^\circx=180∘−148∘=32∘
Final Answer:
x=32∘x = 32^\circx=32∘
Explanation
In every triangle, the three interior angles always add up to a total of 180 degrees. This fact is based on Euclidean geometry and applies to all triangles, whether they are scalene, isosceles, or equilateral.
When two of the angles are given in a triangle, we can find the third angle by subtracting the sum of the two known angles from 180 degrees. This is because the total must remain constant. In this problem, the triangle has two known angles: 42 degrees and 106 degrees. When we add these two together, we get 148 degrees.
To find the value of the unknown angle xxx, we subtract 148 from 180. The result is 32 degrees. This is the measure of the third angle and completes the triangle. This method does not depend on the type or shape of the triangle. It simply relies on the constant property of triangles having interior angles that sum to 180 degrees.
This concept is very useful in geometry problems and also appears in real-world applications like architecture, navigation, and design. It helps in determining unknown measures when working with triangular structures or calculating directions and distances using trigonometry.
