Geometric Inequalities: Chapter Review

Vocabulary and notation

Lengths and angle measures

A side is a segment. In $AB<AC$, the names $AB$ and $AC$ denote lengths. Write an angle measure as $m\angle ABC$, and compare like quantities.

Opposite side

The side that does not touch an angle's vertex. In triangle $ABC$, side $BC$ is opposite angle $A$. Rotation does not change the pairing.

Nondegenerate and degenerate

A nondegenerate triangle has three noncollinear vertices and positive side lengths. A degenerate triangle is a flat boundary arrangement, excluded from ordinary triangle theorems.

Hypotenuse

The side opposite the right angle of a right triangle. The two sides meeting at that angle are its legs.

Perpendicular foot and distance to a line

The perpendicular foot is where a perpendicular from the point meets the line. The distance is that perpendicular length; for a point already on the line, it is zero.

Third-side range

Every possible third-side length when two positive side lengths are fixed. Both geometric boundary values are excluded.

Broken path

A route made of consecutive straight segments. Its travel order matters: points on one line can still create backtracking.

Included angle and matched arms

The included angle is between the two chosen sides, at their common endpoint. In a hinge comparison, matched arms are two pairs of equal corresponding side lengths.

Formulas and theorems

• Side-angle ordering in one triangle

  Each strict comparison below implies the other:

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

  Text description

  Triangle ABC has A at the lower left, B above it, and C at the lower right. Solid side AC is across from the double-marked angle B. Dashed side AB is across from the single-marked angle C. The distinct line and arc patterns identify the pairs; no numerical comparison is given.

  Use when Use one nondegenerate triangle ABC. AC faces angle B; AB faces angle C.

• Equal sides and equal opposite angles

  Each equality below implies the other:

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

  Use when The two sides and their opposite angles belong to the same nondegenerate triangle ABC. These are the isosceles equality cases.

• Exterior angle theorem and inequality

  $$\begin{gathered} e=m\angle A+m\angle B\\ e>m\angle A\\ e>m\angle B\\ e+m\angle C=180^\circ \end{gathered}$$

  Text description

  In triangle ABC, B, C, and D lie in that order on a straight line. A is above the line. The exterior angle e is between CA and CD. The remote interior angles are at A and B; the adjacent interior angle is at C.

  Use when In nondegenerate triangle ABC, extend BC through C to D. Here e is the measure of exterior angle ACD. A and B are remote; C is adjacent.

• Longest side opposite a right or obtuse angle

  In a right triangle, the hypotenuse is uniquely longest. In an obtuse triangle, the side opposite the obtuse angle is uniquely longest.

  Use when Use a nondegenerate triangle and an established right or obtuse angle. Identify its opposite side, regardless of orientation.

• Perpendicular is uniquely shortest

  $$\begin{gathered} PH<PQ \end{gathered}$$

  Text description

  Point P lies above a line through Q, H, and R in that order. PH is perpendicular to the line, with a right-angle square at H. PQ and PR are slanted segments to different endpoints. The right angles identify PH as the perpendicular distance; no numerical lengths are supplied.

  Use when P is outside a line, H is its perpendicular foot, and Q is any other point on that line. If P lies on the line, its distance to the line is zero.

• Triangle inequality

  $$\begin{gathered} a+b>c\\ a+c>b\\ b+c>a \end{gathered}$$

  Use when The positive lengths a, b, and c are sides of a nondegenerate triangle. All three comparisons are strict.

• Positive side-triple and half-perimeter tests

  The proposed lengths form a triangle exactly when either equivalent test holds:

  $$\begin{gathered} a+b>c\\ c<\frac p2 \end{gathered}$$

  Use when The proposed lengths a, b, c are positive and ordered from smallest to largest, allowing ties. Their perimeter is p. Equality is flat; a smaller shorter-side sum cannot close.

• Complete third-side range

  $$\begin{gathered} |a-b|<x<a+b \end{gathered}$$

  Text description

  A number line has an open circle at the difference of the fixed lengths and another at their sum. Only the segment strictly between the circles is highlighted. If the fixed lengths are equal, the difference endpoint is zero and remains excluded.

  Use when The fixed lengths a and b are positive. Every x strictly inside the interval is possible. Equality is a flat or collapsed boundary; when a and b match, the excluded lower endpoint is zero.

• Direct segment versus a broken path

  For any finite broken path, the direct distance is at most the total route length. For three legs:

  $$\begin{gathered} AB\le AP+PQ+QB \end{gathered}$$

  Text description

  Two rows compare a direct segment with a route through P. The first row has A, P, B in that order and travels forward. The second has A, B, P in that order; traveling from A to P and back to B retraces BP. The second route is drawn on offset parallel tracks only to make the repeated travel visible.

  Use when A and B are distinct, and the legs have positive lengths. Equality holds exactly when intermediate points lie on segment AB in travel order, without backtracking. An off-line turn or reversal makes the route longer.

• Polygon inequality

  $$\begin{gathered} s<p-s \end{gathered}$$

  Use when A nondegenerate polygon has vertices not all collinear, perimeter p, and chosen side length s. That side is strictly shorter than all the others together. This necessary bound alone does not construct the polygon.

• Hinge theorem: SAS inequality

  The included-angle comparison implies the third-side comparison:

  $$\begin{gathered} m\angle BAC>m\angle EDF\\ BC>EF \end{gathered}$$

  Text description

  Triangle DEF on the left has a solid outline and the smaller included angle beta at D. Triangle ABC on the right has a dashed outline and the larger included angle alpha at A. Single ticks mark equal lengths DE and AB; double ticks mark equal lengths DF and AC. The opposite side BC is longer than EF. The diagram illustrates the stated hinge comparison.

  Use when Use two nondegenerate triangles ABC and DEF. AB equals DE in length, and AC equals DF in length. The compared angles are between those matched arms.

• Converse hinge theorem

  The third-side comparison implies the included-angle comparison:

  $$\begin{gathered} BC>EF\\ m\angle BAC>m\angle EDF \end{gathered}$$

  Use when Use two nondegenerate triangles ABC and DEF with AB equal to DE and AC equal to DF in length. Reversing the strict side order reverses the included-angle order.

• Hinge equality case

  Equal included angles give equal third sides by SAS. Equal third sides give equal included angles by SSS.

  Use when Both directions require two nondegenerate triangles with AB equal to DE and AC equal to DF in length. Every included angle is positive and less than 180 degrees.

Problem types, step by step

Exam traps

• Trap Choosing an adjacent side or trusting the sketch

  Fix Use the side across from the angle vertex. Orientation and apparent size supply no measurement or incidence.

• Trap Ordering sides in unrelated triangles from angles alone

  Fix Within-triangle ordering needs one triangle. A hinge comparison needs two equal corresponding side pairs, surrounding the compared angles.

• Trap Forcing a unique largest part when two tie

  Fix Equal sides face equal angles. Preserve equalities while transferring the order; compare the proposed largest part with both competitors.

• Trap Comparing an exterior angle with the adjacent angle

  Fix The strict exterior-angle inequality concerns the two remote interior angles. The adjacent interior angle is supplementary and can be larger.

• Trap Calling the shortest sampled segment the distance to a line

  Fix The distance uses the perpendicular foot, compared with every point on the line. Being nearest among several named endpoints does not establish perpendicularity.

• Trap Checking an arbitrary side sum or assuming a variable side is longest

  Fix For positive numerical lengths, test the two shorter against the longest. With variable lengths, justify the order or retain all three inequalities and positivity.

• Trap Dropping the difference bound or including a boundary

  Fix A third side must exceed the absolute difference and stay below the sum. Both bounds are strict; remove endpoints before counting integers.

• Trap Assuming collinearity guarantees an equal-length route

  Fix Equality requires forward travel on the direct segment in the stated order. Backtracking adds distance even on one line.

• Trap Ordering two quantities because both are below the same reference

  Fix That information leaves their mutual order undecided. A transitive comparison needs a chain with a consistent direction.

• Trap Using a known angle outside the matched arms

  Fix The included angle is at the common endpoint of the matched sides. One equal pair or a different labeled angle does not justify a hinge argument.

• Trap Reading exact lengths or every angle from a hinge comparison

  Fix The hinge theorems give the third-side or included-angle order. They do not calculate exact lengths or make every corresponding angle larger.

• Trap Applying the strict triangle inequality to flat auxiliary points

  Fix A construction can give collinear auxiliary points even when the original triangles are nondegenerate. Use the stated point order and segment addition with positive extra pieces, rather than the strict triangle inequality.

• Trap Ruling out only equality in an indirect proof

  Fix Also exclude the reverse strict order, using the earlier forward theorem under its hypotheses. Only then is the desired order the remaining possibility.