R Chandra Solutions for Chapter: Quadrilaterals, Exercise 1: TEST YOUR SKILLS

Two parallel lines $\text{'}\mathrm{l}\text{'}$ and $\text{'}\mathrm{m}\text{'}$ are intersected by a transversal $\mathrm{\text{'}}\mathrm{p}\mathrm{\text{'}}$. Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.

Show that the bisectors of angles of a parallelogram form a rectangle.

In the adjoining figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram in which $\mathrm{\text{'}}\mathrm{P}\mathrm{\text{'}}$ and $\mathrm{\text{'}}\mathrm{Q}\mathrm{\text{'}}$ are midpoints of opposite side $\mathrm{A}\mathrm{B}$ and $\mathrm{C}\mathrm{D}$. If $\mathrm{A}\mathrm{Q}$ intersects $\mathrm{D}\mathrm{P}$ at $\mathrm{S}$ and $\mathrm{B}\mathrm{Q}$ intersects $\mathrm{C}\mathrm{P}$ at $\mathrm{R}$, then show that $\mathrm{P}\mathrm{S}\mathrm{Q}\mathrm{R}$ is parallelogram.

Show that line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.

In a $∆\mathrm{A}\mathrm{B}\mathrm{C}, \mathrm{D}, \mathrm{E}$ and $\mathrm{F}$ are respectively the midpoints of sides $\mathrm{A}\mathrm{B}, \mathrm{B}\mathrm{C}$ and $\mathrm{C}\mathrm{A}$ (see figure). Show that $∆\mathrm{A}\mathrm{B}\mathrm{C}$ is divided into four congruent triangles by joining $\mathrm{D}, \mathrm{E}$ and $\mathrm{F}$.

$\mathrm{l}, \mathrm{m}$ and $\mathrm{n}$ are three parallel lines intersected by transversal $\mathrm{\text{'}}{\mathrm{p}}^{\mathrm{\text{'}}}$and $\mathrm{\text{'}}\mathrm{q}\mathrm{\text{'}}$, such that $\mathrm{l}, \mathrm{m}$ and $\mathrm{}\mathrm{n}$ cut off equal intercepts $\mathrm{A}\mathrm{B}$ and $\mathrm{B}\mathrm{C}$ on $\mathrm{p}$ (see figure). Show that $\mathrm{l}, \mathrm{m}$ and $\mathrm{n}$ cut off equal intercept $\mathrm{D}\mathrm{E}$ and $\mathrm{E}\mathrm{F}$ on $\mathrm{q}$ also.

In the figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a trapezium in which $\mathrm{A}\mathrm{B}\parallel \mathrm{D}\mathrm{C}$ and $\mathrm{E}$ is the midpoint of $\mathrm{A}\mathrm{D}. \mathrm{F}$ is a point on $\mathrm{B}\mathrm{C}$ such that $\mathrm{E}\mathrm{F}\parallel \mathrm{A}\mathrm{B}.$ Prove that $\mathrm{E}\mathrm{F}=\frac{1}{2}(\mathrm{A}\mathrm{B}+\mathrm{D}\mathrm{C})$.

Show that the figure formed by joining the midpoints of the adjacent sides of a quadrilateral is a parallelogram.