Embibe Experts Solutions for Chapter: Congruence of Triangles, Exercise 1: Exercise

Ram drew a right-angle triangle $∆ABC$ such that, $AB=AC$. Then he joined $AD$ such that, $AD\perp BC$.

He asserted, "The base angles, $\angle ACB=\angle ABC$" with the reason that, "$∆ACD\cong ∆DBA$

Which of the following statements is correct?

In the given $∆ABC$, $AB=AC$. $D$ and $E$ are the midpoints of the respective sides. 

If $BE=7 \mathrm{cm}$, then which of these lengths is also $7 \mathrm{cm}$?

Shown below is the rectangle drawn by Sriram. The diagonals of a rectangle are equal and they bisect each other. 

Sriram wants to prove if the given rectangle $ABCD$ can be proved to be a square. Which of this information is the necessary and sufficient condition to prove the same?

In, $∆ABC$, and $∆PQR$, $\angle ABC=\angle QPR$, $AB=PR$ and $BC=PQ$

Which of the following are equal based on the information above?

Which of the following is/are the correct congruence equation for the triangles below?

Three friends, Dhanya, Ekta, and Rashmi are identifying the congruent triangles formed by the diagonals $AC,$ and $BD$ along with sides of the parallelogram $ABCD$. The two diagonals bisect each other.

Dhanya: $\left(i\right)∆ABD\cong ∆CDB \left(ii\right) ∆BAC\cong ∆DCA \left(iii\right)∆AOD\cong ∆COB \left(iv\right)∆COD\cong ∆AOB$

Ekta: $\left(i\right)∆ABD\cong ∆CDB \left(ii\right) ∆BAC\cong ∆DCA \left(iii\right)∆AOD\cong ∆AOB \left(iv\right)∆COD\cong ∆COB$

Rashmi: $\left(i\right)∆ABD\cong ∆ACD \left(ii\right) ∆BAC\cong ∆BDC \left(iii\right)∆AOD\cong ∆COB \left(iv\right)∆COD\cong ∆AOB$

Who identified the congruent triangles correctly?

Jagadish lives near the bank of a river $BC$, and he wants to find the width of the river. He followed the steps below.

$\left(1\right)$ Spotted a tall object at $M$ the other bank $AB$. Marked a point $P$ on $CD$ to get the distance $PM$, to the least possible.

$\left(2\right)$ Chose a point, $X$ on the land. Marked point $O$ on $CD$, such that $MOX$ is a straight line.

$\left(3\right)$ Marked point $R$ on $CD$ such that $OR=OP$

$\left(4\right)$ Marked point $Q$ on $OX$ to get the distance $QR$, to the least possible.

$\left(5\right)$ Measured the distance $RQ$, which he concluded is the width of the river by showing that $∆OPM$ and $∆ORQ$ congruent.

Which of the following rules helped Jagadish to conclude that the triangles are congruent?

Shown below is $∆ABC$ drawn by Gautham. In the triangle, $D$ and $E$ are the midpoints of their respective sides, and $AD$  $BE$ are the altitudes.

What is the measure of $\angle CBE$?