Class 9 Mathematics Triangles Mix Examples - Triangles

Medium

In the given figure,if $PQ = ST$,$QU = TR$,$PQ \perp QT$ and $ST \perp TQ$,then prove that $PR = SU$.

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Solution

(N/A) Given: $PQ = ST$,$QU = TR$,$\angle PQT = 90^\circ$ and $\angle STQ = 90^\circ$.
To prove: $PR = SU$.
Proof:
Consider $\triangle PQR$ and $\triangle STU$.
$1$. $PQ = ST$ (Given)
$2$. $\angle PQR = \angle STU = 90^\circ$ (Given)
$3$. $QR = QU + UR$ and $TU = TR + UR$.
Since $QU = TR$,we have $QR = TR + UR = TU$.
Thus,$QR = TU$.
By $SAS$ congruence criterion,$\triangle PQR \cong \triangle STU$.
Since the triangles are congruent,their corresponding parts are equal $(CPCT)$.
Therefore,$PR = SU$.

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Mix Examples - Triangles

Two lines $l$ and $m$ intersect at the point $O$ and $P$ is a point on a line $n$ passing through the point $O$ such that $P$ is equidistant from $l$ and $m$. Prove that $n$ is the bisector of the angle formed by $l$ and $m$.

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Easy

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In quadrilateral $PQRS$,$PQ = PS$ and $RQ = RS$. Prove that $\angle PQR = \angle PSR$.

Medium

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Medium

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In $\Delta ABC$,$AB = AC$ and $\angle C = 75^{\circ}$,then find $\angle B$. (in $^{\circ}$)

Easy

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In the given figure,if $PQ = ST$,$QU = TR$,$PQ \perp QT$ and $ST \perp TQ$,then prove that $PR = SU$.

Similar Questions

Two lines $l$ and $m$ intersect at the point $O$ and $P$ is a point on a line $n$ passing through the point $O$ such that $P$ is equidistant from $l$ and $m$. Prove that $n$ is the bisector of the angle formed by $l$ and $m$.

In the given figure,$AM$ and $BN$ are both perpendicular to $AB$. $MN$ intersects $AB$ at $P$. Also,$P$ is the midpoint of $AB$. Prove that $AM = BN$ and $P$ is the midpoint of $MN$.

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