Exercises: Congruence from rigid motions

A reflection across line $\ell$ maps $\triangle ABC$ exactly onto $\triangle DEF$. You are given that segment $AB$ maps to segment $DE$, segment $AC$ maps to segment $DF$, and $\angle A$ maps to $\angle D$.

State which triangle congruence criterion (SSS, SAS, ASA, AAS, or HL) can be directly deduced from this rigid motion, and explain why the preservation of distance and angle measure by the reflection guarantees that the two triangles are congruent.

A landscape architect in Seattle is designing a triangular flower bed, $\triangle PQR$. She decides to move the entire design $15 \text{ feet}$ to the east and $5 \text{ feet}$ north to avoid a water pipe, creating a new triangular bed, $\triangle P'Q'R'$.

Since a translation is a rigid motion, all side lengths remain the same. Write a short explanation of how you can use the definition of a translation to prove that $\triangle PQR \cong \triangle P'Q'R'$ using the SSS congruence criterion.

In a graphic design software, a triangular logo $\triangle JKL$ is rotated $90^\circ$ clockwise around the origin to create $\triangle MNO$. The designer knows that $\angle J \cong \angle M$, side $JK \cong MN$, and $\angle K \cong \angle N$.

Identify the congruence criterion that applies to this situation. Briefly explain how the properties of rotations support the use of this specific criterion to conclude that the two logos are identical in size and shape.

You are given two triangles, $\triangle ABC$ and $\triangle XYZ$. A sequence of rigid motions (a translation followed by a reflection) maps $A$ to $X$, $B$ to $Y$, and $C$ to $Z$.

If you know that the rigid motions mapped $\angle B$ to $\angle Y$, $\angle C$ to $\angle Z$, and the side $BC$ to the side $YZ$, which congruence criterion is demonstrated here? State the corresponding parts that align to form this criterion.

Consider a right triangle $\triangle GHI$ with a right angle at $H$. A reflection across the $y$-axis maps $\triangle GHI$ to $\triangle STU$.

If you are given that the hypotenuse $GI$ maps to $SU$, and the leg $GH$ maps to $ST$, state the specific congruence criterion for right triangles that proves $\triangle GHI \cong \triangle STU$. Explain how the rigid motion ensures the necessary conditions for this criterion are met.

A carpenter cuts a triangular piece of wood, $\triangle ABC$, and then uses it as a template to trace a second triangle, $\triangle DEF$, on another wooden board. This tracing process acts as a rigid motion (a translation and a rotation).

If the carpenter measures and confirms that $AB = DE$, $BC = EF$, and $AC = DF$, write a two-step logical argument showing that SSS congruence is a direct consequence of the template being moved as a rigid object.

Complete the missing reason in the following two-column proof snippet:

Given: A rotation about point $P$ maps $\triangle LMN$ to $\triangle RST$.

1. A rotation is a rigid motion. (Reason: Given)

2. Rigid motions preserve angle measures. (Reason: Definition of rigid motion)

3. $\angle L \cong \angle R$ and $\angle M \cong \angle S$. (Reason: ___)

4. Rigid motions preserve segment lengths. (Reason: Definition of rigid motion)

5. Segment $LM \cong Segment RS$. (Reason: Corresponding parts of figures mapped by a rigid motion)

6. $\triangle LMN \cong \triangle RST$. (Reason: ASA Congruence Criterion)

Two triangular metal plates, $\triangle VWX$ and $\triangle YZA$, are produced in a factory in Detroit. A robotic arm picks up $\triangle VWX$, translates it, and places it perfectly on top of $\triangle YZA$.

If the robot's programming ensures that $\angle V$ maps to $\angle Y$, $\angle X$ maps to $\angle A$, and the non-included side $VW$ maps to $YZ$, identify which triangle congruence theorem justifies that the plates are congruent.

An architect in Chicago is designing a symmetrical roof truss, modeled as $\triangle ABC$ with a central support beam $AD$. The design specifications state that $AD \perp BC$ and that $D$ is the midpoint of $BC$.

Consider a reflection of the left triangle, $\triangle ABD$, across the line containing the support beam $AD$. Write a paragraph proof explaining how this reflection maps $\triangle ABD$ onto $\triangle ACD$, and state which congruence criterion (SAS or SSS) is deduced from this rigid motion.

A student claims that if two triangles have two congruent sides and a congruent non-included angle (SSA), the triangles must be congruent.

Using the provided diagram showing a rigid segment $QR$ swinging along an arc to position $QR'$, explain why SSA is a counterexample to triangle congruence. Discuss how the rigid motion of rotation (swinging the segment) creates two distinct triangles, $\triangle PQR$ and $\triangle PQR'$, that share the same SSA measurements.

A surveyor in Texas is mapping two triangular plots of land, $\triangle KLM$ and $\triangle NOP$. A translation maps point $K$ to point $N$. After the translation, a rotation about point $N$ maps ray $KL$ onto ray $NO$.

You are given that $\angle K \cong \angle N$, $\angle M \cong \angle P$, and the side $KL$ is congruent to side $NO$.

Write a two-column proof demonstrating that $\triangle KLM \cong \triangle NOP$. Be sure to state which congruence criterion (ASA or AAS) applies based on the given information.

Copy and complete the following two-column proof.

Given: A sequence of rigid motions maps $\triangle ABC$ onto $\triangle WXY$ such that $AB$ maps to $WX$, $BC$ maps to $XY$, and $AC$ maps to $WY$.

Prove: $\triangle ABC \cong \triangle WXY$

Statements	Reasons
1. A sequence of rigid motions maps $\triangle ABC$ onto $\triangle WXY$.	1. ______
2. Rigid motions preserve distance.	2. Definition of rigid motion
3. $AB \cong WX$, $BC \cong XY$, $AC \cong WY$	3. ______
4. $\triangle ABC \cong \triangle WXY$	4. ______

Two triangular sails, $\triangle DEF$ and $\triangle RST$, are designed for a yacht. The manufacturer states that $\angle D \cong \angle R$, $\angle E \cong \angle S$, and the included side $DE \cong RS$.

Imagine a rigid motion that maps segment $DE$ perfectly onto segment $RS$. Because rigid motions preserve angle measure, explain what must happen to the rays $DF$ and $EF$ during this mapping, and how this logically proves the ASA congruence criterion.

In a mechanical linkage, two triangles, $\triangle GHI$ and $\triangle GJI$, share a common side $GI$. You are given that $GH \cong GJ$ and $HI \cong JI$.

If you were to reflect $\triangle GHI$ across the line containing segment $GI$, describe how the properties of rigid motions and circles (considering $G$ and $I$ as centers of rotation for the segments) prove that point $H$ must map exactly onto point $J$. Conclude which congruence theorem this illustrates.

A utility pole in Florida is supported by two guy wires anchored to the flat ground. The pole forms a right angle with the ground. The two guy wires, representing the hypotenuses of two right triangles, are of equal length ($45 \text{ feet}$). The pole acts as the shared leg for both triangles.

Using the concept of rigid motions (specifically, reflecting one triangle across the utility pole), explain how you can deduce the Hypotenuse-Leg (HL) congruence theorem to prove the two triangles are congruent.

Given $\triangle LMN$ and $\triangle XYZ$ where $LM \cong XY$, $\angle M \cong \angle Y$, and $MN \cong YZ$.

A student performs a translation mapping point $M$ to point $Y$, followed by a rotation mapping ray $ML$ onto ray $YX$. Since $LM \cong XY$, point $L$ maps to point $X$.

Write the next logical steps in this paragraph proof to show that point $N$ must map to point $Z$, and conclude why this sequence of rigid motions proves the SAS congruence criterion.

You are given two triangles on a coordinate plane, $\triangle ABC$ and $\triangle DEF$.

It is known that $AB = DE = 5 \text{ inches}$, $BC = EF = 7 \text{ inches}$, and $\angle B \cong \angle E$.

Construct a formal two-column proof to show that $\triangle ABC \cong \triangle DEF$. Your proof must explicitly use rigid motions: define a translation that maps $B$ to $E$, followed by a rotation about $E$ that maps ray $BA$ to ray $ED$. Justify why point $C$ must map to point $F$ after these motions, establishing the SAS criterion fundamentally.

The Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Consider an isosceles triangle $\triangle PQR$ with $PQ \cong PR$. Let $\ell$ be the angle bisector of $\angle P$.

Write a paragraph proof using rigid motions to prove this theorem. Specifically, analyze a reflection of $\triangle PQR$ across the line $\ell$. Explain where each vertex maps and why, and how this rigid motion proves that $\angle Q \cong \angle R$.

In parallelogram $ABCD$, diagonal $AC$ is drawn. Let $M$ be the midpoint of diagonal $AC$.

Using the properties of a $180^\circ$ rotation about point $M$, prove that $\triangle ABC \cong \triangle CDA$. Your proof must explain how the rotation maps lines to parallel lines, ensuring that segment $AB$ maps exactly to segment $CD$ and segment $BC$ maps exactly to segment $DA$, thereby fulfilling the SSS or ASA congruence criteria through rigid motions.

An engineer is analyzing a bridge truss containing two overlapping triangles, $\triangle WXY$ and $\triangle VYX$, that share the base $XY$.

Given that $\angle WXY \cong \angle VYX$ and $\angle WYX \cong \angle VXY$.

Write a comprehensive two-column proof to show that $\triangle WXY \cong \triangle VYX$. In your reasoning, explicitly state how a reflection of the entire plane across the perpendicular bisector of segment $XY$ maps $\triangle WXY$ to $\triangle VYX$, and identify which standard congruence theorem (ASA, AAS, etc.) this rigid motion establishes.

While SSS, SAS, ASA, and AAS are valid congruence criteria deduced from rigid motions, AAA is not.

Provide a detailed counterexample using a real-world scenario (e.g., a smartphone screen and a flat-screen TV with the same aspect ratio). Explain mathematically why rigid motions (translations, rotations, reflections) cannot map one triangle onto the other in the AAA scenario, and identify which type of transformation would be required, proving that AAA only establishes similarity, not congruence.

The AAS (Angle-Angle-Side) congruence theorem can be proven using the ASA theorem and the Triangle Angle Sum Theorem.

Given $\triangle JKL$ and $\triangle MNO$ with $\angle J \cong \angle M$, $\angle K \cong \angle N$, and non-included side $JL \cong MO$.

Write a multi-step paragraph proof that first uses the Triangle Angle Sum Theorem to show $\angle L \cong \angle O$. Then, describe the specific sequence of rigid motions that will map $\triangle JKL$ onto $\triangle MNO$ using the newly established ASA condition.

A kite $PQRS$ is defined such that $PQ \cong PS$ and $RQ \cong RS$. The main diagonal $PR$ is drawn.

Write a rigorous two-column proof demonstrating that $\triangle PQR \cong \triangle PSR$. Your proof must utilize the definition of a reflection across the line containing $PR$. Explain why points $P$ and $R$ remain fixed, why point $S$ must map to point $Q$, and conclude with the appropriate triangle congruence criterion (SSS) deduced from this transformation.

Consider an equilateral triangle $\triangle ABC$ inscribed in a circle with center $O$.

A rigid motion consisting of a $120^\circ$ clockwise rotation about the center $O$ is performed.

Write an analytical argument explaining how this specific rotation maps the vertices of the triangle onto each other (e.g., $A$ to $B$, $B$ to $C$, $C$ to $A$). Use the properties of this rigid motion to prove that all three sides of the triangle must be congruent ($AB \cong BC \cong CA$), thereby establishing that the triangle is equilateral based on the SSS framework derived from rotational symmetry.