Ex.8.1 Q10 Quadrilaterals Solution - NCERT Maths Class 9

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\(ABCD\) is a parallelogram and \(AP\) and \(CQ\) are perpendiculars from vertices \(A\) and \(C\) on

diagonal \(BD\) (See the given figure). Show that

(i) \(\begin{align} &{ \Delta \mathrm{APB} }{\cong \Delta \mathrm{CQD}}   \end{align}\)

(ii) \(\begin{align} &{ \mathrm{AP}=\mathrm{CQ}}\end{align}\)


 Video Solution
Ex 8.1 | Question 10

Text Solution


What is known?

\(ABCD\) is a parallelogram and \(AP\bot DB,\,CQ\bot DB\)

What is unknown?

How we can show that (i) \(\Delta APB\cong \Delta CQD\)

(ii) \(AP = CQ\)


We can use alternate interior angles and parallelogram property for triangles congruence criterion to show triangles congruent then we can say corresponding parts of congruent triangles will be equal.


(i) In \(\begin{align} \;&\Delta \mathrm{APB} \text { and } \Delta \mathrm{CQD}  \end{align}\)

\[\begin{align}\angle \mathrm{APB}&=\angle \mathrm{CQD}\begin{bmatrix} \text {Each angle} \\ \text{measures } 90^{\circ} \end{bmatrix} \\ {AB}&\!=\!{CD} \begin{bmatrix}\text {Opposite sides of} \\\text{parallelogram } {ABCD}\end{bmatrix} \\ \angle \mathrm{ABP}&\!=\!\angle \mathrm{CDQ} \begin{bmatrix}\text {Alternate interior}\\ \text{angles for AB } \| \mathrm{CD}\!\end{bmatrix}  \\ \therefore \Delta \mathrm{APB} &\cong \Delta \mathrm{CQD}\begin{bmatrix}\text { By AAS }\\\text{congruence rule }\end{bmatrix} \end{align}\]

(ii) By using the above result

\(\begin{align} \triangle \mathrm{APB} \cong \Delta \mathrm{CQD},  \end{align}\) we obtain \(\begin{align}  \mathrm{AP}=\mathrm{CQ}\end{align}\) (By CPCT )

 Video Solution
Ex 8.1 | Question 10