Isosceles Triangle - ICSE STD 9 MATHEMATICS Questions

Q 1[2 Mark Question Answer]2 Marks

$\triangle PQR$ is isosceles with $\text{PQ} = \text{PR}. T$ is the mid$-$point of $\text{QR},$ and $\text{TM}$ and $\text{TN}$ are perpendiculars on $\text{PR}$ and $\text{PQ}$ respectively. Prove that,
$\text{PM} = \text{PN}$

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Q 2[3 marks sum]3 Marks

$\text{DPQ}$ is an isosceles triangle with $DP = DQ$. A straight line $CD$ bisects the exterior $\angle QDR.$ Prove that $DC$ is parallel to $PQ.$

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Q 3[3 marks sum]3 Marks

In the given figure :if $\angle PQS = 60^\circ $,show that $PQ = PS = QS = SR.$

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Q 4[3 marks sum]3 Marks

In $ΔABC, D$ is the mid$-$point of $BC, AD$ is equal to $AC. AC$ is produced to $E$, such that $CE = AC.$ Prove that$:AB = CE$

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Q 5[3 marks sum]3 Marks

In $\triangle ABC, D$ is the mid$-$point of $BC, AD$ is equal to $AC. AC$ is produced to $E,$ such that $CE = AC.$ Prove that:$\angle ADB = \angle DCE$

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Q 6[3 marks sum]3 Marks

In the figure $\triangle ABC$ is isosceles with $AB = AC.$ Prove that:$\angle ADE = \angle BCD$

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Q 7[4 marks sum]4 Marks

In a quadrilateral $\text{PQRS}, PQ = PS$ and $RQ = Rs$. If $\angle 50^\circ $ and $\angle R = 110^\circ $, find $\angle PSR.$

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Q 8[4 marks sum]4 Marks

Prove that the medians corresponding to equal sides of an isosceles triangle are equal.

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Q 9[4 marks sum]4 Marks

The bisector of the equal angles of an isosceles $\triangle PQR$ meet at $O$. If $PQ = PR$, prove that $PO$ bisects $\angle P.$

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Q 10[4 marks sum]4 Marks

Each equal angle of an isosceles triangle is less than the third angle by $15^\circ $. Find the angles.

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Q 11[4 marks sum]4 Marks

In the given figure :if $\angle PQS = 60^\circ ,$find$\angle QPR.$

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Q 12[5 marks sum]5 Marks

Side $BA$ of an isosceles $\triangle ABC$ is produced so that $AB = AD$. If $AB$ and $AC $are the equal sides of the isosceles triangle, prove that $\angle BCD$ is a right angle.