angle-sum-and-classifying-triangles MCQs for UPSC Prelims

26 practice questions on angle-sum-and-classifying-triangles from the A Tale of Three Intersecting Lines section of the UPSC Prelims syllabus. 26 come with a written explanation. Try the sample set below - the answer stays hidden until you ask for it.

7 Easy 13 Medium 6 Hard

Sample questions

Q1
hard

In triangle $ABC$, $\angle C = 80^\circ$ and $\angle A$ is $20^\circ$ more than $\angle B$. What is $\angle B$?

  1. A $60^\circ$
  2. B $30^\circ$
  3. C $40^\circ$
  4. D $50^\circ$
Show answer and explanation

Correct answer: C - $40^\circ$

The angles $\angle A$ and $\angle B$ share $180^\circ - 80^\circ = 100^\circ$. Removing the extra $20^\circ$ leaves $80^\circ$ to be split equally, so $\angle B = 40^\circ$ and $\angle A = 60^\circ$. The value $60^\circ$ is $\angle A$, $50^\circ$ ignores the $20^\circ$ difference, and $30^\circ$ takes $20^\circ$ away from $50^\circ$.

Q2
medium

Which set of angles belongs to a right-angled triangle?

  1. A $60^\circ, 60^\circ, 60^\circ$
  2. B $100^\circ, 40^\circ, 40^\circ$
  3. C $25^\circ, 65^\circ, 90^\circ$
  4. D $45^\circ, 45^\circ, 45^\circ$
Show answer and explanation

Correct answer: C - $25^\circ, 65^\circ, 90^\circ$

The first set contains an angle of exactly $90^\circ$ and adds to $180^\circ$, so it is right-angled. Three angles of $45^\circ$ total only $135^\circ$, so they form no triangle. Three angles of $60^\circ$ give an acute-angled triangle, and the last set is obtuse-angled.

Q3
hard

At one vertex of a triangle, the exterior angle is equal to the interior angle beside it. What follows about the triangle?

  1. A It is equilateral
  2. B It is right-angled at that vertex
  3. C It has two obtuse angles
  4. D No such triangle can be drawn
Show answer and explanation

Correct answer: B - It is right-angled at that vertex

The exterior angle and the interior angle beside it lie on a straight line and add to $180^\circ$, so if they are equal each must be $90^\circ$ and the triangle is right-angled at that vertex. An equilateral triangle would give an interior angle of $60^\circ$ and an exterior angle of $120^\circ$. Two obtuse angles are impossible, and right-angled triangles are common.

Q4
hard

Can a triangle have two obtuse angles?

  1. A Yes, if the third angle is small enough
  2. B No, two angles above $90^\circ$ already add to more than $180^\circ$
  3. C Yes, if the triangle has two equal sides
  4. D No, because every triangle must contain a right angle
Show answer and explanation

Correct answer: B - No, two angles above $90^\circ$ already add to more than $180^\circ$

Each obtuse angle is more than $90^\circ$, so two of them already pass $180^\circ$ before the third angle is counted, which no triangle allows. Even the smallest possible third angle cannot rescue such a triangle. A triangle need not contain a right angle, and equal sides do not change the angle sum.

Q5
hard

Assertion (A): At any vertex of a triangle, the exterior angle and the interior angle beside it add up to $180^\circ$. Reason (R): The three interior angles of a triangle add up to $180^\circ$.

  1. A Both A and R are true, and R is the correct explanation of A
  2. B Both A and R are true, but R is not the correct explanation of A
  3. C A is true, but R is false
  4. D A is false, but R is true
Show answer and explanation

Correct answer: B - Both A and R are true, but R is not the correct explanation of A

The exterior angle and the interior angle beside it sit on one straight line, so they form a linear pair and total $180^\circ$, which makes A true. R is a true property of every triangle. A comes from the straight line at that single vertex rather than from the total of all three interior angles, so R does not explain it.

Q6
medium

What is the reason a triangle cannot contain two right angles?

  1. A Two perpendicular lines cannot be drawn on paper
  2. B Right angles belong only to squares and rectangles
  3. C The third side would be too long to fit on the page
  4. D The two right angles use the whole $180^\circ$, leaving nothing for the third angle
Show answer and explanation

Correct answer: D - The two right angles use the whole $180^\circ$, leaving nothing for the third angle

The angle sum of a triangle is fixed at $180^\circ$, and two right angles already account for all of it, so the third angle would have to be $0^\circ$. Perpendicular lines are easy to draw, page size is not a mathematical limit, and right angles appear in many triangles, though never twice in one.

Q7
easy

Which statement describes an acute-angled triangle?

  1. A One of its angles is exactly $90^\circ$
  2. B One of its angles is more than $90^\circ$
  3. C Each of its three angles is less than $90^\circ$
  4. D Its three sides are all equal
Show answer and explanation

Correct answer: C - Each of its three angles is less than $90^\circ$

An acute-angled triangle needs every one of its three angles below $90^\circ$. An angle of exactly $90^\circ$ makes it right-angled and an angle above $90^\circ$ makes it obtuse-angled. Equal sides describe an equilateral triangle, which is one particular acute-angled triangle rather than the definition.

Q8
easy

Can a triangle have two right angles?

  1. A Yes, if two of its sides are equal
  2. B Yes, if the triangle is drawn large enough
  3. C Yes, if the third angle is very small
  4. D No, such a triangle is impossible
Show answer and explanation

Correct answer: D - No, such a triangle is impossible

Two right angles already use $90^\circ + 90^\circ = 180^\circ$, leaving nothing at all for the third angle, so no such triangle can be drawn. The size of the drawing does not change the angle sum, and equal sides have no effect on it either. Even a very small third angle would push the total past $180^\circ$.

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