Congruent Triangles: Chapter Review

Vocabulary and notation

Congruence and rigid motions

Two figures are congruent when translations, rotations, and reflections map the whole first figure exactly onto the second. Position and orientation may differ; lengths and angle measures do not.

Correspondence and ordered names

In $\triangle ABC\cong\triangle DEF$, positions pair $A$ with $D$, $B$ with $E$, and $C$ with $F$. Side endpoints and an angle’s middle letter identify matching parts.

Included angle and included side

The included angle of two sides is at their common endpoint. The included side of two angles joins their vertices. The terms describe placement, not size.

Isosceles triangle and chosen base

An isosceles triangle has at least two congruent sides. For a chosen pair, their common endpoint is the vertex; the opposite side is the base, and its endpoint angles are the base angles.

Equilateral and equiangular

Equilateral means three congruent sides. Equiangular means three congruent interior angles. Every equilateral triangle is also isosceles under the inclusive definition.

Median, altitude, and internal bisector

A median joins a vertex to the midpoint of the opposite side. An altitude runs from a vertex perpendicularly to the opposite side line. An internal bisector halves the interior angle at its vertex.

Hypotenuse and legs

In a right triangle, the hypotenuse is opposite the right angle. The two legs meet at that angle; identify them from the right-angle location, not the drawing’s orientation.

CPCTC

Corresponding parts of congruent triangles are congruent. This is a consequence of an established triangle match, not a criterion that establishes the match.

Shared part and reflexive equality

A shared segment has the same length in both triangles, such as $AD=AD$. A shared angle requires the same two rays; a shared vertex alone is insufficient.

Auxiliary segment and diagonal

An auxiliary segment helps a proof. A diagonal joins opposite corners of a four-sided figure. Neither supplies unproved angle or length equalities.

Insufficient information

AAA alone or unrestricted SSA alone leaves congruence undecided; neither alone proves noncongruence.

Nondegenerate triangle

A triangle has three distinct, noncollinear vertices and positive side lengths. A collinear arrangement from a failed construction is not a triangle.

Formulas and theorems

• Six corresponding parts

  Congruent triangles have three equal corresponding side lengths and three equal corresponding angle measures. Conversely, all six matches in one correspondence establish congruence.

  Text description

  Triangle ABC and triangle DEF rotated half a turn have three corresponding side pairs and three corresponding angle pairs marked alike.

  Use when Use actual triangles and one consistent vertex matching.

• SSS: Side-Side-Side

  Three equal corresponding side-length pairs establish triangle congruence.

  Text description

  Three matching side pairs have one, two and three ticks. The second triangle is rotated half a turn.

  Use when Start with two actual noncollinear triangles. A proposed list of three positive lengths does not automatically construct a triangle.

  e.g. Two triangles with side lengths $9,16,21$ cm in matching pairs are congruent by SSS.

• SAS: Side-Angle-Side

  Two equal corresponding side-length pairs and congruent included angles establish triangle congruence.

  Text description

  Two separate triangle diagrams distinguish the location of a known angle. In the first, sides AB and AC have side ticks, and the marked angle is at their common vertex A. In the second, sides DE and DF have the same side-tick pattern, but the marked angle is at E. The first angle is included between the two marked sides; the second is not.

  Use when In each triangle the known angle is between the two known sides.

  e.g. Matching sides of $13$ and $22$ cm with matching included $61^\circ$ angles establish SAS.

• ASA: Angle-Side-Angle

  Two congruent corresponding angle pairs and equal included side lengths establish triangle congruence.

  Text description

  Angles at A and D have one arc; angles at B and E have two arcs. Their joining sides AB and DE are marked equal.

  Use when The side joins the two known angle vertices in each triangle, following the same correspondence.

  e.g. Matching endpoint angles $28^\circ,89^\circ$ and the $18$ cm side between them establish ASA.

• AAS: Angle-Angle-Side

  Two congruent corresponding angle pairs and equal corresponding nonincluded side lengths establish triangle congruence.

  Text description

  The angles at A and B match those at D and E. Marked sides AC and DF are not between those angle vertices.

  Use when The given side is not between the two given angles, and the same vertex correspondence holds throughout.

• HL: Hypotenuse-Leg

  Two right triangles are congruent if their hypotenuses and one pair of corresponding legs have equal lengths.

  Text description

  Both triangles have a marked right angle. Their opposite hypotenuses match, and one leg pair matches.

  Use when Both right angles must be stated, marked, or proved. Two arbitrary side pairs without that condition do not supply HL.

  e.g. Two right triangles with hypotenuses $41$ cm and matching legs $9$ cm are congruent by HL.

• Base-angle theorem and its converse

  In triangle ABC, these statements are equivalent:

  $$\begin{gathered} AB=AC\\ m\angle B=m\angle C \end{gathered}$$

  Text description

  In triangle ABC, equal sides AB and AC have one tick each. Their opposite base angles at C and B have matching arcs.

  Use when In nondegenerate triangle ABC, compare each side with its opposite angle. Both directions are valid; the converse is a separate result.

• Isosceles angle calculations

  $$\begin{gathered} b=\frac{180^\circ-v}{2}\\ v=180^\circ-2b \end{gathered}$$

  Text description

  Triangle ABC has equal sides AB and AC, vertex angle v at A, and equal base angles b at B and C.

  Use when Here v is the vertex angle between the chosen equal sides, and b is either equal base angle.

  e.g. For vertex angle $118^\circ$, each base angle is $(180^\circ-118^\circ)/2=31^\circ$.

• Equilateral and equiangular equivalence

  A triangle is equilateral exactly when it is equiangular; every interior angle is 60 degrees. Its three medians are also altitudes and internal bisectors; their lines are its three reflection axes.

  Text description

  All three sides and all three angles are marked equal. Dashed medians join each vertex to the midpoint of its opposite side.

  Use when The side length is not fixed by the angle measures. For an equilateral triangle of perimeter P, each side length is P divided by three.

  e.g. For perimeter $99$ cm, each side is $33$ cm and each angle is $60^\circ$.

• The isosceles vertex segment

  The internal vertex bisector, median, and altitude from the vertex between the chosen equal sides are the same segment. Its line is a reflection axis.

  Text description

  In isosceles ABC, AD bisects the vertex angle. The proved conclusions are shown: equal base halves and a right angle at D.

  Use when The triangle is isosceles and the segment starts at the common endpoint of its equal sides. This coincidence need not hold for a different vertex or an arbitrary triangle.

  e.g. If that segment meets a base of $38$ cm, the base pieces are $19$ cm each and the meeting angles are $90^\circ$.

• AAA and unrestricted SSA

  AAA alone does not fix size. Two side pairs and a nonincluded angle pair, without additional restrictions, do not guarantee congruence.

  Text description

  A schematic ray from A contains two possible points B1 and B2. C is on a second ray. The equal-radius segments from C reach both points on the first ray.

  Use when AAA and SSA can hold for congruent pairs too. SSA constructions may yield zero, one, or two triangles. HL is a valid right-triangle case.

• Construction conditions

  For SAS use two positive lengths and an included angle strictly between zero and 180 degrees. For ASA use a positive base and two positive endpoint angles totaling less than 180 degrees.

  Use when ASA rays open into the same half-plane. A sum of 180 degrees gives parallel rays; a greater sum gives no intersection on the chosen side.

• CPCTC after the match

  An established ordered match

  $$\triangle ABC\cong\triangle DEF$$

  licenses corresponding-part comparisons such as

  $$\begin{gathered} BC=EF\\ \angle ACB\cong\angle DFE \end{gathered}$$

  Text description

  An established ordered match pairs A with D, B with E and C with F. Sides BC and EF and angles at C and F are highlighted as corresponding parts.

  Use when First justify an ordered triangle match using a valid criterion. Never use the target equality as its own premise.

• Transferred lengths in an angle copy

  $$\begin{gathered} OA=PD\\ OB=PE\\ AB=DE\\ \angle AOB\cong\angle DPE \end{gathered}$$

  Text description

  Source OAB and target PDE have recorded radius lengths. The corresponding chords AB and DE have matching double ticks. No angle-equality mark is supplied.

  Use when O and P are source and target vertices; A,B and D,E are radius marks. The source opening is strictly between zero and 180 degrees; the radius is positive. The angle match follows from SSS, then CPCTC.

Problem types, step by step

Exam traps

• Trap Matching letters by the drawing’s positions

  Fix Use the ordered correspondence. Rotations and reflections can change every left-right visual cue.

• Trap Confusing side equality with triangle congruence

  Fix Use $=$ for equal lengths or measures and $\cong$ for congruent segments, angles, or triangles. State a criterion for a whole-triangle match.

• Trap Calling any two sides and any angle SAS

  Fix Locate the common endpoint of the two known sides. The angle at that endpoint is the included one.

• Trap Ignoring the side’s endpoints in ASA or AAS

  Fix Match the angle vertices first. Their joining side is included; another corresponding side is nonincluded.

• Trap Using HL because a triangle looks right

  Fix Establish a right angle in each triangle and identify the side opposite it before checking the hypotenuse and leg pairs. Two equal leg pairs and the included right angle give SAS.

• Trap Treating insufficient data as proof of noncongruence

  Fix AAA or unrestricted SSA alone leaves congruence undecided. An unequal corresponding side or angle can instead rule out a stated match.

• Trap Doubling the vertex angle instead of the base angle

  Fix An isosceles triangle has two equal base angles and one vertex angle. Identify those roles from the equal-side markings before using the angle sum.

• Trap Excluding equilateral triangles from isosceles

  Fix Isosceles means at least two equal sides. An equilateral triangle allows any vertex to play the equal-side vertex role.

• Trap Giving every median or altitude three roles

  Fix The shared roles require an isosceles triangle and its segment from the vertex between the chosen equal sides.

• Trap Using the target equality as evidence for its own proof

  Fix Build the triangle congruence from other justified data. CPCTC comes afterward, including separately at each stage of a two-stage proof.

• Trap Inventing facts after drawing a segment or separating triangles

  Fix Preserve the original givens and supply reasons for every new equality. An auxiliary segment alone supplies neither bisection nor perpendicularity.

• Trap Assuming a compass copy already has the desired angle

  Fix Use the transferred radii and chord for SSS; the angle equality is the CPCTC conclusion, not an SAS premise.

• Trap Extending SSS to a four-sided frame

  Fix Four fixed side lengths need not fix the angles. SSS applies to triangles; a diagonal brace adds a side condition to each triangular piece.