Class 9 Mathematics Triangles Textbook - Triangles

Medium

$AD$ is an altitude of an isosceles triangle $ABC$ in which $AB = AC$. Show that:
$(i)$ $AD$ bisects $BC$
$(ii)$ $AD$ bisects $\angle A$.

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Solution

(N/A) $(i)$ In $\Delta ABD$ and $\Delta ACD$,we have:
$AB = AC$ [Given]
$\angle ADB = \angle ADC = 90^\circ$ [Since $AD$ is an altitude]
$AD = AD$ [Common side]
Therefore,by $RHS$ congruence rule,$\Delta ABD \cong \Delta ACD$.
Since the triangles are congruent,their corresponding parts are equal $(CPCT)$.
Therefore,$BD = CD$,which means $AD$ bisects $BC$.
$(ii)$ Since $\Delta ABD \cong \Delta ACD$,their corresponding angles are equal $(CPCT)$.
Therefore,$\angle BAD = \angle CAD$,which means $AD$ bisects $\angle A$.

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Textbook - Triangles

$l$ and $m$ are two parallel lines intersected by another pair of parallel lines $p$ and $q$ (see Fig). Show that $\Delta ABC \cong \Delta CDA$.

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In an isosceles triangle $ABC$ with $AB = AC$,$D$ and $E$ are points on $BC$ such that $BE = CD$ (see Fig). Show that $AD = AE$.

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$AB$ and $CD$ are respectively the smallest and longest sides of a quadrilateral $ABCD$ (see Fig). Show that $\angle A > \angle C$ and $\angle B > \angle D$.

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Two sides $AB$ and $BC$ and median $AM$ of one triangle $ABC$ are respectively equal to sides $PQ$ and $QR$ and median $PN$ of $\Delta PQR$ (see Fig). Show that:
$(i)$ $\Delta ABM \cong \Delta PQN$
$(ii)$ $\Delta ABC \cong \Delta PQR$

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Line-segment $AB$ is parallel to another line-segment $CD$. $O$ is the mid-point of $AD$ (see Fig). Show that $(i)$ $\Delta AOB \cong \Delta DOC$ $(ii)$ $O$ is also the mid-point of $BC$.

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$AD$ is an altitude of an isosceles triangle $ABC$ in which $AB = AC$. Show that:$(i)$ $AD$ bisects $BC$$(ii)$ $AD$ bisects $\angle A$.

Similar Questions

$l$ and $m$ are two parallel lines intersected by another pair of parallel lines $p$ and $q$ (see Fig). Show that $\Delta ABC \cong \Delta CDA$.

In an isosceles triangle $ABC$ with $AB = AC$,$D$ and $E$ are points on $BC$ such that $BE = CD$ (see Fig). Show that $AD = AE$.

$AB$ and $CD$ are respectively the smallest and longest sides of a quadrilateral $ABCD$ (see Fig). Show that $\angle A > \angle C$ and $\angle B > \angle D$.

Two sides $AB$ and $BC$ and median $AM$ of one triangle $ABC$ are respectively equal to sides $PQ$ and $QR$ and median $PN$ of $\Delta PQR$ (see Fig). Show that:$(i)$ $\Delta ABM \cong \Delta PQN$$(ii)$ $\Delta ABC \cong \Delta PQR$

Line-segment $AB$ is parallel to another line-segment $CD$. $O$ is the mid-point of $AD$ (see Fig). Show that $(i)$ $\Delta AOB \cong \Delta DOC$ $(ii)$ $O$ is also the mid-point of $BC$.

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$AD$ is an altitude of an isosceles triangle $ABC$ in which $AB = AC$. Show that:
$(i)$ $AD$ bisects $BC$
$(ii)$ $AD$ bisects $\angle A$.

Two sides $AB$ and $BC$ and median $AM$ of one triangle $ABC$ are respectively equal to sides $PQ$ and $QR$ and median $PN$ of $\Delta PQR$ (see Fig). Show that:
$(i)$ $\Delta ABM \cong \Delta PQN$
$(ii)$ $\Delta ABC \cong \Delta PQR$