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Difficulty: EasyTriangle Properties and Angle Theorems

The measures of the three interior angles of a triangle are in the ratio 2:3:52:3:5. What is the measure, in degrees, of the largest angle of the triangle?

  1. A
    18∘18^\circ
  2. B
    36∘36^\circ
  3. C
    50∘50^\circ
  4. D
    54∘54^\circ
  5. 90∘90^\circAnswer

Answer

90 degrees
The sum of the measures of the interior angles of a triangle is 180∘180^\circ. Given the ratio 2:3:52:3:5, the angles can be expressed as 2x2x, 3x3x, and 5x5x. Adding these yields 2x+3x+5x=1802x + 3x + 5x = 180, which simplifies to 10x=18010x = 180. Solving for xx gives x=18x = 18. The largest angle is represented by 5x5x. Substituting 1818 for xx gives 5×18=905 \times 18 = 90. Therefore, the measure of the largest angle is 90∘90^\circ.

Step-by-Step Solution

1
Set up an equation representing the sum of the angles in a triangle.
2x+3x+5x=1802x + 3x + 5x = 180
The sum of the measures of the interior angles of any triangle is always 180∘180^\circ. We can represent the angles as 2x2x, 3x3x, and 5x5x using the given ratio.
2
Solve for the scale factor xx.
10x=180⇒x=1810x = 180 \Rightarrow x = 18
Combine like terms to find the total number of parts, then divide 180180 by 1010 to find the value of one part.
3
Calculate the measure of the largest angle.
5×18=905 \times 18 = 90
The largest angle corresponds to the largest term in the ratio, which is 5x5x. Multiplying the scale factor 1818 by 55 gives the measure of the largest angle.

Key Concept

The interior angles of a triangle sum to 180∘180^\circ. Ratios can be solved by defining a common multiplier for each part and setting their sum equal to the total.
Estimated Time:45s
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Topics

Triangle Properties and Angle Theorems

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