Name: Parallel & Transversal Lines: Illustration
Uploaded: 2019-07-19
Duration: 6 min 20 s
Description: In geometry, the straight lines which do not meet are called parallel. A line that crosses a pair of parallel lines is known as transversal. Watch and learn...

Question

Prove that a triangle must have at least two acute angles.

Accepted Answer

Given: $∆ A B C$ is a triangle.

To Prove: $∆ A B C$ must have at least two acute angles.

Proof: Let us assume that $∆ A B C$ has only one acute angle i.e. less than $90 °$ .
Then the other two angles will have to be right or obtuse angles (i.e. $90 °$ or more than $90 °$ ).
Case 1: Let us assume that both the angles are right angles.

And, Let the third angle be 'x', which is an acute angle.

By angle sum property of triangle, we have
$x + 90 ° + 90 ° = 180 ° x = 0 °$
As we know that an angle of a triangle can never be $0 °$ . So, our assumption is completely wrong that two angles of a triangle are right angles. Therefore, this case is not possible.

Now, if we consider other cases i.e, out of two angles, if one is $90 °$ and other is obtuse or both angles are obtuse angles, then by angle sum property of triangle, measure of third angle will be negative, which is impossible.

Hence, our assumption that $∆ A B C$ has only one acute angle is wrong.

Therefore, $∆ A B C$ must have atleast two acute angles. Hence, Proved.

Prove that a triangle must have at least two acute angles.

Important Questions on Lines and Angles

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