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Congruence, Proof, and Constructions

Unit 1 of Geometry. 14 questions below, each with the working. Every answer was checked by a second pass before it was published.

Transformations in the plane, rigid motions, congruence, triangle congruence criteria, proofs about lines, angles, triangles and parallelograms, constructions.

How this unit is tested

Start by getting the transformation rules cold: translations, reflections over the axes and over y=x, and rotations about the origin by 90°, 180°, and 270°. Practice applying these as coordinate rules until you can do them without hesitation, then connect them to the bigger idea: a rigid motion (isometry) is any transformation that preserves distance and angle measure, and two figures are congruent exactly when some sequence of rigid motions maps one onto the other. Once the transformational definition of congruence is solid, move to the triangle congruence criteria: SSS, SAS, ASA, AAS, and HL (right triangles only). For each problem, identify which parts are marked congruent and check that the pattern matches a valid criterion — especially whether an angle is 'included' between two given sides. Learn to recognize and reject AAA and SSA, since these come up constantly as distractor answers. For proof problems, work in order: state what's given, mark up the diagram, find the theorem that connects the given information to what you need (vertical angles, linear pairs, parallel-line angle relationships, the Isosceles Triangle Theorem, or parallelogram properties), then chain those into a two-column or paragraph proof that ends with CPCTC when you need to transfer a congruence from triangles to their corresponding parts. Finally, practice constructions physically or by visualizing the compass arcs: perpendicular bisector, angle bisector, copying a segment or angle, and constructing parallel or perpendicular lines through a point. Exam questions often ask you to justify why a construction works, which always comes back to the compass creating equal radii, hence equidistant points.

What you have to know

Rigid Motion (Isometry)
A transformation is a rigid motion if it preserves distance between every pair of points and preserves angle measure; translations, rotations, and reflections are rigid motions, but dilations (scale factor ≠ 1) are not.
Definition of Congruence
Two figures are congruent if and only if there exists a sequence of rigid motions that maps one figure exactly onto the other.
SAS Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.
ASA and AAS Congruence Theorems
If two angles and the included side of one triangle are congruent to the corresponding parts of another (ASA), or two angles and a non-included side are congruent (AAS), the triangles are congruent.
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, each pair of alternate interior angles is congruent; conversely, if a pair of alternate interior angles is congruent, the lines are parallel.
Parallelogram Diagonal Theorem
The diagonals of a parallelogram bisect each other, and either diagonal divides the parallelogram into two congruent triangles.

14 practice questions

  1. Which transformation below is NOT a rigid motion?
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    Answer. A dilation with scale factor 3

    Dilations change the size of a figure (unless the scale factor is 1), so they don't preserve distance; reflections, rotations, and translations all preserve distance and angle measure, making them rigid motions (isometries).
  2. Find the image of the point $(3,-2)$ under a reflection over the line $y=x$.
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    Answer. $(-2,3)$

    Reflecting over $y=x$ swaps the coordinates: $(x,y)\to(y,x)$. Applying this to $(3,-2)$ gives $(-2,3)$.
  3. Find the image of $(2,5)$ under a $90^\circ$ counterclockwise rotation about the origin.
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    Answer. $(-5,2)$

    The rule for a $90^\circ$ counterclockwise rotation about the origin is $(x,y)\to(-y,x)$. Applying it to $(2,5)$ gives $(-5,2)$.
  4. According to the transformational definition used in Common Core geometry, what does it mean for two figures to be congruent?
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    Answer. Two figures are congruent if and only if there exists a sequence of rigid motions (reflections, rotations, translations) that maps one figure exactly onto the other.

    This definition replaces the older 'same size and shape' idea with a precise transformational one: since rigid motions preserve distance and angle measure, a figure and its image under such a sequence have exactly matching corresponding parts.
  5. In triangles ABC and DEF, $AB \cong DE$, $\angle A \cong \angle D$, and $AC \cong DF$. Which postulate proves $\triangle ABC \cong \triangle DEF$, and why must the angle be positioned as it is?
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    Answer. SAS (Side-Angle-Side); the congruent angle must be the angle included between the two congruent sides.

    SAS requires the marked angle to lie between the two given sides. Here angle A is between sides AB and AC (and angle D is between DE and DF), so the included-angle condition is satisfied and SAS applies.
  6. Right triangles ABC and DEF each have a right angle at C and F. The hypotenuse AB ≅ DE, and leg BC ≅ EF. Which congruence criterion applies, and why doesn't ordinary SSA justify it instead?
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    Answer. HL (Hypotenuse-Leg); HL is a special case that only works for right triangles because the right angle fixes the triangle's shape, unlike the general ambiguous SSA case.

    SSA is not valid for general triangles because two different triangles can share two sides and a non-included angle. HL is an exception that works only when the known angle is the right angle, since that pins the third side uniquely.
  7. Two parallel lines are cut by a transversal. One angle measures $128^\circ$. Find the measures of its alternate interior angle and its same-side (co-interior) interior angle.
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    Answer. Alternate interior angle = $128^\circ$; same-side interior angle = $52^\circ$.

    Alternate interior angles formed by a transversal crossing parallel lines are congruent, so that angle is also $128^\circ$. Same-side interior angles are supplementary, so $180^\circ - 128^\circ = 52^\circ$.
  8. In triangle ABC, $\angle A = 50^\circ$ and $\angle B = 70^\circ$. Find the measure of the exterior angle at vertex C.
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    Answer. $120^\circ$

    By the Exterior Angle Theorem, an exterior angle of a triangle equals the sum of the two remote interior angles: $50^\circ+70^\circ=120^\circ$. (The interior angle at C is $180^\circ-120^\circ=60^\circ$, confirming the triangle angle sum.)
  9. An isosceles triangle has a vertex angle of $36^\circ$. Find the measure of each base angle.
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    Answer. $72^\circ$ each

    The base angles of an isosceles triangle are congruent, and all three angles sum to $180^\circ$. So each base angle is $(180^\circ-36^\circ)/2 = 72^\circ$.
  10. Outline a proof that the base angles of isosceles triangle ABC (with AB ≅ AC) are congruent.
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    Answer. Draw the angle bisector from vertex A to side BC, meeting it at point D. Then triangle ABD ≅ triangle ACD by SAS (AB≅AC, angle BAD≅angle CAD by construction, AD≅AD by reflexive property), so angle B ≅ angle C by CPCTC.

    This is the classic proof of the Isosceles Triangle Theorem. The auxiliary segment AD (the bisector of the vertex angle) creates two triangles that can be shown congruent by SAS, and CPCTC then gives the base angles congruent.
  11. Parallelogram ABCD has vertices $A(0,0)$, $B(4,0)$, $C(6,3)$, $D(2,3)$. Show that the diagonals bisect each other.
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    Answer. Midpoint of AC = $(3,1.5)$ and midpoint of BD = $(3,1.5)$; since both diagonals share the same midpoint, they bisect each other.

    Using the midpoint formula, midpoint of AC is $\left(\frac{0+6}{2},\frac{0+3}{2}\right)=(3,1.5)$ and midpoint of BD is $\left(\frac{4+2}{2},\frac{0+3}{2}\right)=(3,1.5)$. Equal midpoints prove the diagonals cross at the same point.
  12. In parallelogram ABCD, prove that $\angle A \cong \angle C$.
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    Answer. Draw diagonal BD. By the Alternate Interior Angles Theorem, angle ABD ≅ angle CDB and angle ADB ≅ angle CBD (since AB∥CD and AD∥BC). With BD≅BD, triangle ABD ≅ triangle CDB by ASA, so angle A ≅ angle C by CPCTC.

    Splitting the parallelogram with a diagonal produces two congruent triangles by ASA using pairs of alternate interior angles and the shared diagonal; CPCTC then transfers the congruence to the non-adjacent angles A and C.
  13. Describe the compass-and-straightedge steps to construct the perpendicular bisector of segment AB.
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    Answer. Open the compass to more than half of AB's length. With the point on A, draw arcs above and below the segment; with the same radius, put the point on B and draw arcs that cross the first two. Draw a line through the two intersection points; this line is the perpendicular bisector of AB.

    Because the compass radius is the same from both A and B, every point on the constructed line is equidistant from A and B, which is exactly the defining property of the perpendicular bisector.
  14. Describe the compass-and-straightedge steps to bisect a given angle, angle ABC.
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    Answer. With the compass point at vertex B, draw an arc that crosses both rays, marking points D and E. Using the same fixed radius, place the compass at D and draw an arc inside the angle, then repeat from E with the same radius so the arcs intersect at point F. Ray BF is the bisector of angle ABC.

    Points D and E are equidistant from B, and F is equidistant from D and E, which by symmetry makes BF split angle ABC into two congruent angles.

What people get wrong

  1. Treating AAA (or just AA) as a congruence criterion — it only proves similarity, not congruence, because it says nothing about size. Only use SSS, SAS, ASA, AAS, or HL to prove triangles congruent.
  2. Using SSA as if it were valid — two sides and a non-included angle can produce two different triangles (the 'ambiguous case'). If the given angle is not between the two given sides, look for HL (right triangle only) or find another pair of parts instead.
  3. Mixing up rotation directions — writing $(x,y)\to(y,-x)$ when a 90° counterclockwise rotation was asked for. Memorize the two rules separately: CCW 90° is $(x,y)\to(-y,x)$, CW 90° is $(x,y)\to(y,-x)$, and check with a simple point like $(1,0)$ if unsure.
  4. Writing proof steps without citing a definition, postulate, or theorem for each one — a correct-looking chain with missing justifications loses credit. State the reason (e.g., 'Alternate Interior Angles Theorem,' 'Reflexive Property') next to every step.
  5. Confusing alternate interior angles (congruent) with same-side interior angles (supplementary) when parallel lines are cut by a transversal. Sketch the transversal and label all eight angles before deciding which relationship applies.
  6. Assuming a parallelogram's diagonals are congruent or perpendicular without that being given — those properties only hold for special parallelograms (rectangles and rhombi, respectively). Stick to the general parallelogram properties (opposite sides/angles congruent, diagonals bisect each other) unless told otherwise.

Drill this unit until it sticks

These questions come back on a schedule built from what you get wrong, alongside the rest of Geometry. Free, and no account needed to start.

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