In a triangle, if an exterior angle is 69° and its opposite interior angles are x and 2x°, find the sizes of each interior angle of the triangle.
Answer (2)
The interior angles of the triangle are 23°, 46°, and 111°, calculated using the Exterior Angle Theorem and the Triangle Sum Theorem. The exterior angle of 69° was used to find the opposite angles, represented as x and 2x. By solving the equations, we determined the values of all three angles.
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Use the exterior angle theorem: x + 2 x = 69 .
Solve for x : x = 23 .
Calculate the second angle: 2 x = 46 .
Find the third angle using the triangle angle sum theorem: y = 180 − 23 − 46 = 111 . The three interior angles are 23° , 46° , 111° .
Explanation
Problem Analysis Let's analyze the problem. We are given a triangle with an exterior angle of 69°. The two non-adjacent interior angles are x and 2 x . We need to find the measure of each interior angle of the triangle.
Apply Exterior Angle Theorem The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. Therefore, we can write the equation: x + 2 x = 69
Simplify the equation Combine like terms: 3 x = 69
Solve for x Divide both sides by 3 to solve for x :
x = 3 69 = 23
Calculate the angles x and 2x Now we know that one interior angle is x = 23° , and another is 2 x = 2 ( 23 ) = 46° .
Use the Triangle Angle Sum Theorem The sum of the interior angles of a triangle is 180°. Let the third interior angle be y . Then: x + 2 x + y = 180
Substitute the values of x and 2x Substitute the values of x and 2 x :
23 + 46 + y = 180
Simplify the equation Simplify: 69 + y = 180
Solve for y Subtract 69 from both sides to solve for y :
y = 180 − 69 = 111
State the final answer Therefore, the three interior angles of the triangle are 23° , 46° , and 111° .
Examples
Understanding triangle angles is crucial in architecture. When designing roofs, the angles must be precise to ensure stability and proper water runoff. For example, knowing the exterior angle helps determine the interior angles needed for the roof's structure to be sound and functional, preventing collapses and water damage.
