Question

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 __________.

Answer 1

Define vertically opposite angles with figure

A pair of vertical angles formed at the opposite side of a vertex due to intersection of two lines are called vertically opposite angles

Vertically opposite angles are always equal

From the figure we can see that

$\angle m,\angle n$ and $\angle b,\angle a$ are pairs of vertically opposite angles formed by the intersection of the lines $PQ$ and $RS$

$\Rightarrow$$\angle m=\angle n$ and $\angle b=\angle a$

From the figure we can also observe that $\angle m,\angle n$ are pair of acute angles

Consider $\angle b$ and $\angle m$

These angles form the line $RS$

The angles forming a line are supplementary angles

$\Rightarrow$$\angle b+\angle m={180}^{\circ }$

$\Rightarrow$ $\angle b={180}^{\circ }-\angle m$

As $\angle m$ is acute we know that, $\angle m<{90}^{\circ }$

$\Rightarrow$ $\angle b>{90}^{\circ }$

Hence, $\angle b,\angle a$ are pair of obtuse angles.

Hence, it is proven that if one pair of vertically opposite angles is acute then the other pair of vertically opposite angles must be obtuse, when two lines intersect each other.