Examkin

Question

Difficulty: MediumTriangle Properties and Angle Theorems

In a certain triangle, the ratio of the measure of the first angle to the measure of the second angle is 1:21:2. The measure of the third angle is 20∘20^\circ less than the measure of the second angle. What is the measure of the largest angle in the triangle?

  1. A
    40∘40^\circ
  2. B
    64∘64^\circ
  3. 80∘80^\circAnswer
  4. D
    100∘100^\circ
  5. E
    120∘120^\circ

Answer

80∘80^\circ
The correct answer is 80∘80^\circ. By representing the first angle as xx, the second angle is 2x2x, and the third angle is 2x−202x - 20. The sum of the interior angles in a triangle is always 180∘180^\circ, which gives the equation x+2x+(2x−20)=180x + 2x + (2x - 20) = 180. Simplifying this results in 5x−20=1805x - 20 = 180, which yields 5x=2005x = 200 and x=40x = 40. Substituting this value back into the expressions for the three angles gives measures of 40∘40^\circ, 80∘80^\circ, and 60∘60^\circ. Comparing these values shows that the largest angle is 80∘80^\circ.

Step-by-Step Solution

1
Define the measures of the first and second angles using a single variable based on their ratio.
Let the first angle be xx and the second angle be 2x2x.
Since the ratio of the first angle to the second angle is 1:21:2, we can represent them as xx and 2x2x respectively.
2
Express the measure of the third angle in terms of the same variable.
The third angle is 2x−202x - 20.
The problem states the third angle is 20∘20^\circ less than the second angle, which has a measure of 2x2x.
3
Set up an equation using the triangle angle sum theorem and solve for xx.
x+2x+(2x−20)=180  ⟹  5x−20=180  ⟹  5x=200  ⟹  x=40x + 2x + (2x - 20) = 180 \implies 5x - 20 = 180 \implies 5x = 200 \implies x = 40.
The sum of the measures of the interior angles of any triangle is always 180∘180^\circ.
4
Calculate the measures of all three angles to determine which is the largest.
First angle = 40∘40^\circ, Second angle = 2(40∘)=80∘2(40^\circ) = 80^\circ, Third angle = 2(40∘)−20∘=60∘2(40^\circ) - 20^\circ = 60^\circ. The largest angle is 80∘80^\circ.
We must substitute x=40x = 40 back into our expressions for each angle to find their actual degree measures and identify the largest one.

Key Concept

The interior angles of a triangle always sum to 180∘180^\circ. Ratios and word problems can be modeled algebraically to determine unknown angle measures.
Estimated Time:1m 15s
Rate this question

Topics

Triangle Properties and Angle Theorems

More questions from this topic