Triangle ABC has the angle measures shown. Measure of angle A = (2 x) degrees. Measure of angle B = (3 x) degrees. Measure of angle C = (4 x) degrees. Which statement is true about the angles? Measure of angle A = 20 degrees Measure of angle B = 60 degrees Angle A and Angle B are complementary Measure of angle A + measure of angle C = 100 degrees
The Correct Answer and Explanation is:
Let’s break this down step by step.
We are given that:
- Measure of angle A = 2x
- Measure of angle B = 3x
- Measure of angle C = 4x
In any triangle, the sum of the angles must always be 180 degrees. Therefore, we can set up the following equation:Angle A+Angle B+Angle C=180∘\text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circAngle A+Angle B+Angle C=180∘
Substituting the expressions for the angles in terms of xxx:2x+3x+4x=1802x + 3x + 4x = 1802x+3x+4x=180
Simplifying the left side:9x=1809x = 1809x=180
Now, solving for xxx:x=1809=20x = \frac{180}{9} = 20x=9180=20
Now that we know x=20x = 20x=20, we can find the measures of the individual angles:
- Measure of angle A = 2x=2(20)=40∘2x = 2(20) = 40^\circ2x=2(20)=40∘
- Measure of angle B = 3x=3(20)=60∘3x = 3(20) = 60^\circ3x=3(20)=60∘
- Measure of angle C = 4x=4(20)=80∘4x = 4(20) = 80^\circ4x=4(20)=80∘
Now, let’s go through the given options:
- Measure of angle A = 20 degrees: This is incorrect because we found that angle A is 40 degrees, not 20 degrees.
- Measure of angle B = 60 degrees: This is correct since we calculated that angle B is 60 degrees.
- Angle A and Angle B are complementary: Complementary angles add up to 90 degrees. But angle A is 40 degrees, and angle B is 60 degrees. Together, they add up to 100 degrees, not 90, so this is false.
- Measure of angle A + measure of angle C = 100 degrees: This is correct because angle A is 40 degrees, and angle C is 80 degrees, so 40+80=10040 + 80 = 10040+80=100 degrees.
Conclusion:
The correct statements are:
- Measure of angle B = 60 degrees
- Measure of angle A + measure of angle C = 100 degrees
These two statements are true.
