Triangles - Congruence Theorems

If $\triangle ABC \cong \triangle PQR$ under the correspondence $ABC \leftrightarrow PQR$, which of the following is true by CPCT?

In the context of structural stability, 'rigidity' means that a shape cannot be deformed without changing the length of its sides. Which shape possesses this inherent property?

In an isosceles triangle $ABC$ used in a roof gable, if $AB = AC$ and $AD$ is the altitude to $BC$, then $\triangle ABD \cong \triangle ACD$ by which rule?

A 'Warren Truss' bridge uses a series of equilateral triangles. If all triangles in the truss have sides of 5 meters, which property ensures all these triangles are congruent?

If two altitudes of a triangle are equal, then the triangle is isosceles. This can be proven by which congruence rule applied to the right triangles formed by the altitudes?

In a structural brace, $\triangle PQR$ has $PQ = PR$. $S$ is a point on $QR$ such that $PS$ is the altitude. If $QS = 3\text{ cm}$, find the length of $QR$.

To prove that the diagonals of a rectangle $ABCD$ are equal, we can show that $\triangle ABC \cong \triangle BAD$. Which congruence criterion is used?

A foldable tent pole is held in place by two guy-wires $SA$ and $SB$ anchored at $A$ and $B$. The pole $SO$ is perpendicular to the ground $AB$. If the wires are of equal length ($SA = SB$), the pole is stable. Which congruence rule proves the pole is exactly in the middle of the anchors (i.e., $AO = OB$)?

In a transmission tower, two triangles $\triangle ABC$ and $\triangle DBC$ stand on the same base $BC$ and on the same side of it. If $AB = DC$ and $AC = DB$, which congruence criterion ensures the tower's symmetrical rigidity by proving $\triangle ABC \cong \triangle DCB$?

Two support struts $OA$ and $OB$ are bolted to a wall at points $A$ and $B$. A third strut $OC$ is attached such that $OC$ bisects $\angle AOB$. If the wall is vertical and $A$ and $B$ are equidistant from the level of $O$, which congruence theorem proves that the tension in $OA$ and $OB$ is balanced by making $\triangle OAC \cong \triangle OBC$ (assuming $AC = BC$)?

In the given figure, $AD$ is the perpendicular bisector of $BC$. Which congruence criterion can be used to prove that $\triangle ABD \cong \triangle ACD$?

In a triangle, if two angles and a non-included side are equal to corresponding parts of another triangle, the triangles are congruent by:

If $\triangle ABC \cong \triangle FDE$, then which side of $\triangle FDE$ is equal to $BC$?

In the given figure, $AB = CD$ and $AB \parallel CD$. Which criterion proves $\triangle AOB \cong \triangle DOC$?

In the given figure, if $AB = DC$ and $AD = BC$, then $\triangle ABC \cong \triangle CDA$ by which criterion?

If $\triangle ABC \cong \triangle PQR$ by ASA, and $\angle A = \angle P$, $\angle B = \angle Q$, then which side must be equal?

To prove that the diagonals of a rectangle are equal, which triangles should we prove congruent?

In a geometric proof for a square $ABCD$ with diagonals intersecting at $O$, to show that $OA = OB = OC = OD$, we first prove $\triangle AOB \cong \triangle BOC$. Given $AB = BC$, $BO$ is common, and $\angle ABO = \angle CBO = 45^\circ$, which rule is used?

In a compass-and-straightedge construction of an angle bisector of $\angle BAC$, points $D$ and $E$ are marked on $AB$ and $AC$ such that $AD = AE$. Arcs of equal radii from $D$ and $E$ intersect at $F$. Which criterion proves $\triangle ADF \cong \triangle AEF$?

Two right-angled triangular plates are manufactured such that their hypotenuses are equal and one pair of corresponding acute angles is equal. By which rule are the two triangles congruent?

In the given figure, $ABC$ is a triangle in which the altitude $AD$ bisects the side $BC$. Which congruence criterion can be used to prove that $\triangle ABD \cong \triangle ACD$?

In $\triangle ABC$, if $BC = AB$ and $\angle B = 80^\circ$, then $\angle A$ is equal to:

In $\triangle ABC$ and $\triangle XYZ$, $\angle A = \angle X$, $\angle B = \angle Y$ and $AB = XY$. Which rule proves they are congruent?

In an equilateral triangle $ABC$, each interior angle measures:

If $\triangle ABC \cong \triangle PQR$, and the perimeter of $\triangle ABC$ is $25$ cm, what is the perimeter of $\triangle PQR$?

In $\triangle ABC$, if $AB = AC$ and $AD$ is the median to $BC$, then which congruence rule most directly proves $\triangle ABD \cong \triangle ACD$?

Which of the following conditions is sufficient to prove that a triangle is equilateral?

A student is given two triangles $\triangle ABC$ and $\triangle PQR$ where $AB = PQ$, $AC = PR$ and altitude $AD =$ altitude $PS$. To prove $\triangle ABC \cong \triangle PQR$, which step is required first?

If $\triangle ABC$ is an equilateral triangle and $D, E, F$ are midpoints of $BC, CA, AB$ respectively, why is $\triangle DEF$ also equilateral?

In $\triangle ABC$, the bisectors of $\angle B$ and $\angle C$ intersect at $O$. If a line is drawn through $O$ parallel to $BC$ intersecting $AB$ at $D$ and $AC$ at $E$, prove the relationship between $DE$, $BD$, and $CE$.

In right triangles $ABC$ and $DEF$, if hypotenuse $AC = DF$ and side $AB = DE$, then $\triangle ABC \cong \triangle DEF$ by which rule?

In $\triangle ABC$, if $BC = AB$ and $\angle B = 80^\circ$, then $\angle A$ is:

If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent by:

In the given figure, $AB = AC$ and $AD$ is the bisector of $\angle BAC$. Which of the following is true by $CPCT$?

In the given figure, $ABCD$ is a quadrilateral in which $AD = BC$ and $\angle DAB = \angle CBA$. To prove that $\triangle ABD \cong \triangle BAC$, which congruence criterion is used, and what is the relationship between $\angle ABD$ and $\angle BAC$?

In $\triangle ABC$, if $\angle A = 40^\circ$ and $\angle B = 70^\circ$, then the triangle is:

In the figure, $BA \perp AC$ and $DE \perp DF$ such that $AB = DE$ and $BF = CD$. To prove $\triangle ABC \cong \triangle DEF$ by $RHS$, we must show that:

In an isosceles $\triangle ABC$ with $AB = AC$, $D$ and $E$ are points on $BC$ such that $BE = CD$. Is $\triangle ABD \cong \triangle ACE$? If so, by which rule?

Given two right-angled triangles $\triangle ABC$ and $\triangle DEF$, where $\angle B = \angle E = 90^\circ$. If the hypotenuse $AC = DF$ and one side $BC = EF$, they are congruent by RHS. If instead we are given $AB = DE$ and $BC = EF$, which criterion applies?

A surveyor is measuring a plot $ABCD$ where $AB \parallel CD$ and $AB = CD$. He connects the diagonals $AC$ and $BD$. If they intersect at $O$, which congruence theorem is used to prove $\triangle AOB \cong \triangle COD$?