Criteria for Congruence of Triangles

Prove, $△ABD\cong △ACD$. Also, prove that $\angle B=\angle C$ and $BD=CD$.

Which of the following statements is/are not true for congruency of triangles.

$\left(i\right)$Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle.

$\left(ii\right)$If all the corresponding angles of two triangles are equal, then the triangles are congruent.

$\left(iii\right)$If two sides and a non-included angle of a triangle are equal to the two corresponding sides and a corresponding non-included angle of other triangle then the triangles are congruent.

In $△PQR$, $QX$ and $RX$ are bisectors of $\angle PQR$ and $\angle PRQ$ respectively. If $XS$ and $XT$ are perpendiculars from $X$ on $QR$ and $PQ$ respectively, prove that $PX$ bisects the angle $P$

$△ABC$ is right-angled at $B$; $ACDE$ and $BCGF$ are squares. Prove that $BD=AG$.

The image of an object placed at a point $A$ before a plane mirror $LM$ is seen at the point $B$ by an observer at $D$ as in the adjoining figure. Prove that the image is as far behind the mirror as the object is in front of the mirror.

 In the given figure, $AC=BC, \angle DCA=\angle ECB$ and $\angle DBC=\angle EAC$. Prove that $DC=EC$.

If the above two isosceles triangles are congruent and if $\angle B=110°$, then find $\angle B_1$.

If the above two isosceles triangles are congruent and if $\angle B=120°$, then find $\angle B_1$.

If the equal sides and the angle included between the equal sides of the triangles are equal to each other, then $\frac{AC}{DF}=1$.

If the equal sides and the angle included between the equal sides of the triangles are equal to each other, then prove that $AC=DF$.

If the equal sides and the angle included between the equal sides of the triangles are equal to each other, then prove that $\angle A=\angle D$.

Prove that any two isosceles triangles are congruent if the equal sides and the angle included between the equal sides of the triangles are equal to each other?

If their sides are equal to each other in the two right-angled isosceles triangles given, then $\angle A=\angle E$. 

In the two right-angled isosceles triangles, prove that $\angle B=\angle E$ if their sides (other than the hypotenuse) are equal to each other?

In the two right-angled isosceles triangles, prove that $AB=DE$ if their sides (other than the hypotenuse) are equal to each other?

Prove that any two right-angled isosceles triangles are congruent if their sides are equal to each other?

What is congruence among the right-angled isosceles triangles?

State True or False:

Two equilateral triangles are always congruent.

Sheela drew a triangle whose dimensions were x cm, y cm and z cm.

Meena drew a triangle whose dimensions were exactly half that of Sheela’ triangle.

Leela said both triangles followed the SSS rule of congruency as they were exactly proportional.

Reena pointed out that they were similar and not congruent.

Tina gave a solution to make the triangles congruent.

Pick the right option.

Which value of x will prove ∆ ABC ≌ ∆ DEF by SSS postulate?