NCERT Solutions for Chapter: Lines and Angles, Exercise 4: Exercise

If two lines intersect, prove that the vertically opposite angles are equal.

Bisectors of interior $\angle B$ and exterior $\angle ACD$ of a $△ABD$ intersect at the point $T$. Prove that

$\angle \mathrm{BTC}=\frac{1}{2}\angle \mathrm{BAC}$.

A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angles so formed are parallel.

Prove that through a given point, we can draw only one perpendicular to a given line.

Prove that two lines that are respectively perpendicular to two intersecting lines intersect each other.
[Hint: Use proof by contradiction]

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

In the given fig., $\angle \mathrm{Q}>\angle \mathrm{R}, \mathrm{PA}$ is the bisector of $\angle \mathrm{QPR}$ and  $PM\perp QR$. Prove that $\angle \mathrm{APM}=\frac{1}{2}(\angle \mathrm{Q}-\angle \mathrm{R})$.