If two lines intersect at a point and if one pair of vertically opposite angles are always

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If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are _______.

Concept: Related Angles - Concept of Vertically Opposite Angles

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We can see that line OP and QR intersect at point S where angle OSR and angle QSP is considered as acute angles denoted by $\theta $ . Now, we know the definition of linear pair angle i.e. two angles are said to be linear if they are adjacent angles formed by two intersecting lines. The measure of this linear pair angle is $180{}^\circ $ . So, in our figure, we will find the value of linear pair angle given as below:

Here, $\angle OSQ+\angle QSP=180{}^\circ $ which is denoted by $\alpha $ . So, on solving this equation and finding angle $\angle OSQ$ , we get$\angle OSQ=180-\angle QSP$ ……………………..(1)But it is already given to us that $\angle QSP$ is acute angle which means angle less than 90. So, suppose if we take maximum value of acute angle i.e. 89, then we get value as $\angle OSQ=180-89=91$ which is the value of obtuse angle. So, another angle will be greater than 90 which is known as obtuse angle. Hence, option (a) is correct.Note: Remember to consider the value of acute angles which is always less than 90. Sometimes in a bit of a hurry students consider $90{}^\circ $ as an acute angle and on solving they do not get value of another angle greater than $90{}^\circ $. Also, do not consider that if one pair of vertically opposite angles is an acute angle then other pairs of vertically opposite angles will be acute angles only. This is a misconception students have. So, do not make this mistake.

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If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are Obtuse angles.

An angle which is greater than 90° and less than 180° is called obtuse angle. In other words an obtuse angle is between a right angle and a straight angle.

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