DAV Class 7 Maths Chapter 9 Worksheet 3

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DAV Class 7 Maths Chapter 9 Worksheet 3

Congruent Triangles Worksheet 3


1. In the figure, \( \angle P = \angle Y = 40^o \text{ and } \angle Q = \angle Z = 60^o \). Find the third pair of corresponding parts which make \( \triangle \text{PQR} \cong \triangle \text{YZX} \) by ASA congruence condition.

Solution

\[ \begin{align*} \text{In } \triangle PQR & \text{ and } \triangle YZX \\ \angle \text{P} &= \angle \text{Y} \implies 40^\circ \text{ (Given angle A)} \\ \angle \text{Q} &= \angle \text{Z} \implies 60^\circ \text{ (Given angle A)} \\ \\ \text{Third pair} & \text{ of corresponding parts is } \\ \text{PQ} &= \text{YZ} \implies 4.6 \,cm \text{ (Included Side S)} \\ \\ \text{By ASA} & \text{ congruence condition,} \\ & \triangle PQR \cong \triangle YZX \end{align*} \]

Answer Third pair of corresponding parts is \( \color{red} \text{PQ} = \text{YZ} \)

2. Which pairs of triangles are congruent by ASA congruence condition in the given figure? If congruent, write the congruence of the two triangles in symbolic form.

(i)

Solution

\[ \begin{align*} \text{In } \triangle ABC & \text{ and } \triangle XYZ \\ \angle \text{A} &= \angle \text{Z} \implies 40^\circ \text{ (Given) }\color{green} A\\ \text{AC} &= \text{ZY} \implies 3.5 \,cm \text{ (Given) }\color{green} S\\ \angle \text{C} &= \angle \text{Y} \implies 60^\circ \text{ (Given) } \color{green} A \\ \\ \text{By ASA} & \text{ congruence condition,} \\ & \triangle ABC \cong \triangle ZXY \end{align*} \]

Answer \( \color{red} \triangle ABC \cong \triangle ZXY \)

(ii)

Solution

\[ \begin{align*} \text{In } \triangle ABO & \text{ and } \triangle CDO \\ \angle \text{ABO} &= \angle \text{CDO} \implies 80^\circ \text{ (Given) }\color{green} A\\ \text{BO} &= \text{DO} \implies 4 \,cm \text{ (Given) }\color{green} S\\ \angle \text{AOB} &= \angle \text{COD} \implies \text{ (Vertically Opposite Angles) } \color{green} A \\ \\ \text{By ASA} & \text{ congruence condition,} \\ & \triangle AOB \cong \triangle COD \end{align*} \]

Answer \( \color{red} \triangle AOB \cong \triangle COD \)

(iii)

Solution

\[ \begin{align*} \text{In } \triangle ABC & \text{ and } \triangle DCB \\ \angle \text{ABC} &= \angle \text{DCB} \implies 40^\circ + 20^\circ = 60^\circ\text{ (Given) }\color{green} A\\ \text{BC} &= \text{CB} \implies \text{ (Common Side) }\color{green} S\\ \angle \text{ACB} &= \angle \text{DBC} \implies 40^\circ \text{ (Given) } \color{green} A \\ \\ \text{By ASA} & \text{ congruence condition,} \\ & \triangle ABC \cong \triangle DCB \end{align*} \]

Answer \( \color{red} \triangle ABC \cong \triangle DCB \)

(iv)

Solution

\[ \begin{align*} \text{In } \triangle ABC & \text{ and } \triangle AED \\ \angle \text{B} &= \angle \text{E} \implies 80^\circ \text{ (Angle sum property) } \color{green} A \\ \text{BC} &= \text{ED} \implies 4 \,cm \text{ (Given) }\color{green} S\\ \angle \text{C} &= \angle \text{D} \implies 60^\circ \text{ (Given) }\color{green} A\\ \\ \text{By ASA} & \text{ congruence condition,} \\ & \triangle ABC \cong \triangle AED \end{align*} \]

Answer \( \color{red} \triangle ABC \cong \triangle AED \)

(v)

Solution

\( \text{In } \triangle ABC \text{ and } \triangle XZY \), sides of the triangle are not equal. So the triangles are not congruent.

Answer \( \color{red} \text{Not congruent} \)

3. In the given figure, QX bisects \( \angle PQR \) as well as \( \angle PSR \). State the three facts needed to ensure that \( \triangle QRS \cong \triangle QPS \). Give reasons for each statement.

Solution

\[ \begin{align*} \color{magenta} Given : & \text{ QX bisects } \angle PQR \text{ and } \angle PSR \\ \color{magenta} \text{To prove} : & \, \, \triangle QRS \cong \triangle QPS \\ \\ \text{In } \triangle QPS & \text{ and } \triangle QRS \\ \angle \text{PQS} &= \angle \text{RQS} \implies \text{(Given) }\color{green} A \\ \text{QS} &= \text{QS} \implies \text{(Common side) }\color{green} S\\ \angle \text{PSQ} &= \angle \text{RSQ} \implies \text{(Given) } \color{green} A \\ \\ \text{By ASA} & \text{ congruence condition,} \\ & \triangle QRS \cong \triangle QPS \end{align*} \]

4. In the given figure, PS bisects \( \angle P \) and PS \( \perp \) QR.

(i) Find the three pairs of matching parts to check whether \( \triangle PSQ \cong \triangle PSR \) or not.

Answer

\[ \begin{align*} \color{magenta} Given : & \text{ PS bisects } \angle P \text{ and } PS \perp QR \\ \\\text{In } \triangle PSQ & \text{ and } \triangle PSR \\ \angle \text{QPS} &= \angle \text{RPS} \implies \text{(Given) }\color{green} A \\ \text{PS} &= \text{PS} \implies \text{(Common side) }\color{green} S\\ \angle \text{PSQ} &= \angle \text{PSR} \implies \text{(Given) } \color{green} A \\ \\ \text{By ASA} & \text{ congruence condition,} \\ & \triangle PSQ \cong \triangle PSR \end{align*} \]

(ii) Is \( \triangle PSQ \cong \triangle PSR \) ?

Answer Yes , by ASA congruence condition.

(iii) Is it true to say that QS = SR? Why?

Answer Yes QS = SR, by corresponding parts of congruent triangles (C.P.C.T).

5. In the given figure, \( AO = BO \) and \( \angle A = \angle B. \)

(i) Is \( \angle AOC = \angle BOD \) ? Why?

Answer Yes \( \color{red} \angle AOC = \angle BOD \). Vertically opposite angles.

(ii) Is \( \triangle AOC \cong \triangle BOD \) by ASA congruence condition ?

Answer

\[ \begin{align*} \color{magenta} Given & : AO = BO \text{ and } \angle A = \angle B \\ \color{magenta} \text{To prove} & : \triangle AOC \cong \triangle BOD \\ \\\text{In } \triangle PSQ & \text{ and } \triangle PSR \\ \angle A &= \angle B \implies \text{(Given) }\color{green} A \\ \text{AB} &= \text{BO} \implies \text{(Given) }\color{green} S\\ \angle AOC &= \angle BOD \implies \text{(Vertically opposite angles) } \color{green} A \\ \\\text{By ASA} & \text{ congruence condition,} \\ & \triangle AOC \cong \triangle BOD \end{align*} \]

(iii) Is \( \angle ACO = \angle BDO \) ? Why?

Answer Yes \( \angle ACO = \angle BDO \), by corresponding parts of congruent triangles (C.P.C.T).