In a triangle, if one angle is twice the size of another angle, what is the meas

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Question: In a triangle, if one angle is twice the size of another angle, what is the measure of the smallest angle? (2023)

Options:

Correct Answer: 30 degrees

Exam Year: 2023

Solution:

Let the smallest angle be x. Then the second angle is 2x, and the third angle is 180 - 3x. Solving gives x = 30 degrees.

In a triangle, if one angle is twice the size of another angle, what is the measure of the smallest angle? (2023)

In a triangle, if one angle is twice the size of another angle, what is the measure of the smallest angle? (2023)

Step 1: Understand that in a triangle, the sum of all angles is always 180 degrees.
Step 2: Let the smallest angle be represented as 'x'.
Step 3: Since one angle is twice the size of another, let the second angle be '2x'.
Step 4: The third angle can be found by subtracting the sum of the first two angles from 180 degrees. So, the third angle is '180 - (x + 2x)'.
Step 5: Simplify the expression for the third angle: it becomes '180 - 3x'.
Step 6: Now, we have three angles: x, 2x, and 180 - 3x.
Step 7: Since all angles must be positive, set up the equation: x + 2x + (180 - 3x) = 180.
Step 8: Simplify the equation: 180 = 180, which is always true, so we need to find a specific value for x.
Step 9: To find the smallest angle, we can assume that the angles must be positive. Therefore, we can set '3x < 180' to find the maximum value for x.
Step 10: Solve for x: 3x < 180 gives x < 60 degrees.
Step 11: Since we know that the smallest angle is x, we can try values for x that are less than 60 degrees.
Step 12: Testing x = 30 degrees: the second angle would be 2x = 60 degrees, and the third angle would be 180 - 3x = 60 degrees.
Step 13: All angles (30, 60, 60) are positive and add up to 180 degrees, confirming that x = 30 degrees is a valid solution.

Triangle Angle Sum – The sum of the angles in a triangle is always 180 degrees.
Variable Representation – Using variables to represent unknown angles helps in setting up equations.
Proportional Relationships – Understanding the relationship between angles, such as one being twice another.