Congruence of Triangles

If $\angle A \cong \angle B$ and $m\angle A = 75^\circ$, then what is the measure of $\angle B$?

Which of the following describes the relationship between a figure and its carbon copy?

Two rectangles are congruent if they have the same ________.

In the symbol $\cong$, the top part '~' represents ________.

Two triangles are congruent if two angles and a non-included side of one are equal to the corresponding angles and side of another. This is known as the ________ criterion.

In $\triangle PQR$, if $\angle Q = 90^\circ$, $PQ = 3$ cm, and $QR = 4$ cm. If $\triangle PQR \cong \triangle XYZ$, what is the length of $XZ$?

If $\triangle ABC \cong \triangle LMN$, and the perimeter of $\triangle ABC$ is 24 cm, what is the perimeter of $\triangle LMN$?

Two triangles are drawn on a coordinate plane. $\triangle ABC$ has vertices $A(0,0), B(4,0), C(0,3)$ and $\triangle PQR$ has vertices $P(5,5), Q(9,5), R(5,8)$. Are these triangles congruent?

If $\triangle ABC \cong \triangle FDE$, which of the following represents the correct correspondence of vertices?

In a geometric design, two right-angled triangles $\triangle ABC$ and $\triangle DEF$ have hypotenuses of equal length ($13$ cm) and one pair of corresponding legs equal ($5$ cm). Which theorem ensures they can be perfectly overlaid?

Two line segments are $12\text{ cm}$ each. Are they congruent?

If $\triangle ABC \cong \triangle FED$ under the correspondence $ABC \leftrightarrow FED$, which angle corresponds to $\angle B$?

In the context of congruence, the symbol '$\\cong$' denotes:

If line segment $PQ \cong RS$, and $RS \cong TU$, then which of the following is true?

If the perimeter of a regular pentagon is $25\text{ cm}$, and it is congruent to another regular pentagon, what is the length of one side of the second pentagon?

A circle with radius $r$ is congruent to a circle with area $49\pi$. What is the value of $r$?

If two line segments $AB$ and $CD$ are congruent, and $AB$ is doubled in length to become $A'B'$, while $CD$ is tripled to become $C'D'$, what is the ratio of $A'B'$ to $C'D'$?

A logo designer creates two overlapping circular discs, $C1$ and $C2$. For the discs to be congruent, their radii must be identical. Disc $C1$ has a radius of $r_1 = \frac{2y + 8}{4}$ units, and Disc $C2$ has a radius of $r_2 = \frac{y + 10}{3}$ units. Determine the value of $y$ that makes the figures congruent, and calculate the diameter of one such disc.

An architect is designing two decorative window frames. Frame A consists of two line segments, $PQ$ and $RS$, where $PQ = (3x - 5)$ cm and $RS = (x + 7)$ cm. If these two line segments are congruent, find the value of $x$. Also, if Frame B is a rectangle composed of four such segments of length $PQ$ as its sides, what would be the total length of the wood required for the perimeter of Frame B?

Segments $MN$ and $XY$ are congruent. $MN$ is the diameter of a circle with area $49\pi$ sq units. What is the length of segment $XY$?

If $\triangle ABC \cong \triangle PQR$, then which of the following is true?

Which of the following is NOT a criterion for congruence of triangles?

If $\triangle ABC$ and $\triangle DEF$ are such that $AB = DE$, $BC = EF$, and $\angle B = \angle E = 90^\circ$, they are congruent by:

In $\triangle ABC$, if $AB = AC$ and $AD \perp BC$, which criterion is most directly used to show $\triangle ABD \cong \triangle ACD$ using the right angle?

If $\triangle ABC \cong \triangle PQR$ by SAS criterion, and $AB=PQ$, $BC=QR$, then which angle must be equal?

In the figure, $AC = BD$ and $AD = BC$. Which statement is true?

Two triangles are congruent by the ASA criterion if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other. In $\triangle ABC$ and $\triangle DEF$, if $\angle A = \angle D$ and $\angle B = \angle E$, which side must be equal?

Two triangles are formed by the diagonal of a rectangle $PQRS$. If the diagonal is $PR$, which criterion best proves $\triangle PQR \cong \triangle RSP$ using only the basic properties of a rectangle (opposite sides equal and interior angles are $90^{\circ}$)?

In a circle with center $O$, two chords $AB$ and $CD$ are of equal length. If we join the ends of the chords to the center to form $\triangle OAB$ and $\triangle OCD$, which criterion proves $\triangle OAB \cong \triangle OCD$?

Given two triangles $\triangle LMN$ and $\triangle OPQ$. If $LM = OP$, $MN = PQ$, and $\angle N = \angle Q$, why is this NOT enough to prove congruence in all cases?