A K Upadhyay and S Upadhyay Solutions for Chapter: Quadrilaterals, Exercise 4: TEST YOURSELF

In a quadrilateral $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$, the angles, $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$ are in the ratio $1:2:3:4$. Find the measure of each angles of the quadrilateral.

In a quadrilateral $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$, $\mathrm{A}\mathrm{O}$ and $\mathrm{B}\mathrm{O}$ are bisectors of $\mathrm{\angle }\mathrm{A}$ and $\mathrm{\angle }\mathrm{B}$ respectively. Prove that $\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{B}=\frac{1}{2}(\mathrm{\angle }\mathrm{C}+\mathrm{\angle }\mathrm{D})$

Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a rhombus with $\mathrm{\angle }\mathrm{A}\mathrm{B}\mathrm{C}={56}^{\mathrm{o}}$. Determine $\mathrm{\angle }\mathrm{A}\mathrm{C}\mathrm{D}$

$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a trapezium in which $\mathrm{A}\mathrm{B}\parallel \mathrm{D}\mathrm{C},\mathrm{B}\mathrm{D}$ is a diagonal and $\mathrm{E}$ the midpoint of $\mathrm{A}\mathrm{D}$. A line is drawn through $\mathrm{E}$ parallel to $\mathrm{A}\mathrm{B}$ intersecting $\mathrm{B}\mathrm{C}$ at $\mathrm{F}$. Show that $F$ is the midpoint of $\mathrm{B}\mathrm{C}$.

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Prove that the line segment joining the mid-point of the diagonals of a trapezium is parallel to each of the parallel sides and is equal to half the difference of these sides.

Two parallel lines $l$ and $m$ are intersected by a transversal $p$. Show that the quadrilaterial formed by the bisectors of interior angles is a rectangle.