幾何與角

In the figure, $\widehat{AED}=1$ and $\widehat{CFD}=4$. If $\angle ABC=100^\circ$, then $\angle ABD=$

How many different triangles can be constructed so that the lengths of the three sides are $x\text{ cm}$, $2x\text{ cm}$ and $12\text{ cm}$, where $x$ is an integer?

In the figure, $ABCD$ is a parallelogram. $E$ and $F$ are points on $AD$ and $BC$ respectively such that $AB/\!/EF$. $EF$ meets $AC$ at $G$. If $AG:GC=1:2$, then area of $ABFG$ : area of $EGCD=$

In the figure, $E$ and $F$ are the mid-points of $AB$ and $AC$ respectively. $G$ and $H$ are points on $BD$ and $CD$ respectively such that $\dfrac{DG}{GB}=\dfrac{DH}{HC}=\dfrac{3}{5}$. If $EF=6\text{ cm}$, then $GH=$

In the figure, $EAF$ is a common tangent to the circles at the point $A$. Chords $AC$ and $BC$ of the smaller circle are produced to meet the larger circle at $G$ and $D$ respectively. Which of the following must be true? I. $\angle ADG=\angle EAG$ II. $\angle ABD=\angle AGD$ III. $\angle BAE=\angle ADB$

In the figure, $AEC$ is a diameter and $DEB$ is a straight line. Find $x$.

Which of the following pairs of triangles is/are similar?

In the figure, $x=$

In the figure, $O$ is the centre of the circle, $AOB$ is a straight line and $BCD$ is the tangent to the circle at $C$. Find $x$.

In the figure, $\widehat{AB}=\widehat{BC}=\dfrac{1}{2}\widehat{CD}$. Find $\angle ABC$.