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The smallest angle of a triangle is equal to two thirds of the smallest angle of a quadrilateral. The ratio between the angles of the quadrilateral is 3: 4: 5: 6. The largest angle of the triangle is twice its smallest angle. What is the sum, in degrees, of the second largest angle of the triangle and the largest angle of the quadrilateral?
- a)50 °
- b)55 °
- c)180 °
- d)25 °
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The smallest angle of a triangle is equal to two thirds of the smalle...
Let the angles of quadrilateral are 3x, 4x, 5x, 6x
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So, 3x + 4x + 5x + 6x = 360
x = 20
Smallest angle of quadrilateral = 3 × 20 = 60 °
Smallest angle of triangle = 2/3 ×60∘ = 40∘
Largest angle of triangle = 2 × 40 ° = 60 °
Three angles of triangle are 40 °, 60 °, 80 °
Largest angle of quadrilateral is 120 °
Sum (2nd largest angle of triangle + largest angle of quadrilateral)
= 60 ° + 120 ° = 180 °.
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The smallest angle of a triangle is equal to two thirds of the smalle...
Understanding the Problem
To solve the problem, we need to analyze the relationships between the angles of the triangle and the quadrilateral.
Angles of the Quadrilateral
- The ratio of the angles in the quadrilateral is 3:4:5:6.
- Let the angles be represented as \(3x\), \(4x\), \(5x\), and \(6x\).
- The sum of angles in a quadrilateral is \(360°\):
\[
3x + 4x + 5x + 6x = 360
\]
- Simplifying gives \(18x = 360\), thus \(x = 20\).
- The angles are:
- \(3x = 60°\)
- \(4x = 80°\)
- \(5x = 100°\)
- \(6x = 120°\)
Angles of the Triangle
- Let the smallest angle of the triangle be \(T_s\).
- From the problem, \(T_s = \frac{2}{3}(smallest \, angle \, of \, quadrilateral)\):
\[
T_s = \frac{2}{3} \times 60° = 40°
\]
- The largest angle of the triangle \(T_l\) is twice the smallest angle:
\[
T_l = 2 \times T_s = 2 \times 40° = 80°
\]
- The sum of the angles in a triangle is \(180°\). Let the second largest angle \(T_{sl}\) be:
\[
T_s + T_{sl} + T_l = 180°
\]
- Substituting known values:
\[
40° + T_{sl} + 80° = 180°
\]
\[
T_{sl} = 180° - 120° = 60°
\]
Finding the Required Sum
- The second largest angle of the triangle \(T_{sl} = 60°\).
- The largest angle of the quadrilateral is \(120°\).
- Therefore, the sum is:
\[
T_{sl} + (largest \, angle \, of \, quadrilateral) = 60° + 120° = 180°
\]
Thus, the correct answer is option 'C', which is \(180°\).
To solve the problem, we need to analyze the relationships between the angles of the triangle and the quadrilateral.
Angles of the Quadrilateral
- The ratio of the angles in the quadrilateral is 3:4:5:6.
- Let the angles be represented as \(3x\), \(4x\), \(5x\), and \(6x\).
- The sum of angles in a quadrilateral is \(360°\):
\[
3x + 4x + 5x + 6x = 360
\]
- Simplifying gives \(18x = 360\), thus \(x = 20\).
- The angles are:
- \(3x = 60°\)
- \(4x = 80°\)
- \(5x = 100°\)
- \(6x = 120°\)
Angles of the Triangle
- Let the smallest angle of the triangle be \(T_s\).
- From the problem, \(T_s = \frac{2}{3}(smallest \, angle \, of \, quadrilateral)\):
\[
T_s = \frac{2}{3} \times 60° = 40°
\]
- The largest angle of the triangle \(T_l\) is twice the smallest angle:
\[
T_l = 2 \times T_s = 2 \times 40° = 80°
\]
- The sum of the angles in a triangle is \(180°\). Let the second largest angle \(T_{sl}\) be:
\[
T_s + T_{sl} + T_l = 180°
\]
- Substituting known values:
\[
40° + T_{sl} + 80° = 180°
\]
\[
T_{sl} = 180° - 120° = 60°
\]
Finding the Required Sum
- The second largest angle of the triangle \(T_{sl} = 60°\).
- The largest angle of the quadrilateral is \(120°\).
- Therefore, the sum is:
\[
T_{sl} + (largest \, angle \, of \, quadrilateral) = 60° + 120° = 180°
\]
Thus, the correct answer is option 'C', which is \(180°\).
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The smallest angle of a triangle is equal to two thirds of the smallest angle of a quadrilateral. The ratio between the angles of the quadrilateral is 3: 4: 5: 6. The largest angle of the triangle is twice its smallest angle. What is the sum, in degrees, of the second largest angle of the triangle and the largest angle of the quadrilateral?a)50 °b)55 °c)180 °d)25 °Correct answer is option 'C'. Can you explain this answer?
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The smallest angle of a triangle is equal to two thirds of the smallest angle of a quadrilateral. The ratio between the angles of the quadrilateral is 3: 4: 5: 6. The largest angle of the triangle is twice its smallest angle. What is the sum, in degrees, of the second largest angle of the triangle and the largest angle of the quadrilateral?a)50 °b)55 °c)180 °d)25 °Correct answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The smallest angle of a triangle is equal to two thirds of the smallest angle of a quadrilateral. The ratio between the angles of the quadrilateral is 3: 4: 5: 6. The largest angle of the triangle is twice its smallest angle. What is the sum, in degrees, of the second largest angle of the triangle and the largest angle of the quadrilateral?a)50 °b)55 °c)180 °d)25 °Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The smallest angle of a triangle is equal to two thirds of the smallest angle of a quadrilateral. The ratio between the angles of the quadrilateral is 3: 4: 5: 6. The largest angle of the triangle is twice its smallest angle. What is the sum, in degrees, of the second largest angle of the triangle and the largest angle of the quadrilateral?a)50 °b)55 °c)180 °d)25 °Correct answer is option 'C'. Can you explain this answer?.
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