Angles and Parallel Lines: Chapter Review

Vocabulary and notation

Angle objects and measures

$\angle AOB$ names an angle; $m\angle AOB$ names its measure. Keep the vertex in the middle. Congruent angles have equal measures. Named openings are non-reflex unless specified otherwise.

Adjacent angles and linear pairs

Adjacent angles share a vertex and a side, with nonoverlapping interiors. A linear pair is adjacent with opposite noncommon rays.

Complementary and supplementary angles

Complementary measures total $90^\circ$; supplementary measures total $180^\circ$. Neither relationship requires adjacency.

Vertical angles

Opposite sectors formed by two distinct intersecting lines; their sides occur in opposite-ray pairs.

Parallel lines and a transversal

Parallel lines are distinct coplanar lines that never meet. A transversal crosses two coplanar lines at distinct points. Interior angles lie between the crossed lines; exterior angles lie outside that strip.

Corresponding angles

Angles in the same relative position at the two crossings of one transversal.

Alternate and same-side angles

Alternate interior or exterior pairs lie at two different crossings, on opposite sides of one transversal. Same-side interior or exterior pairs lie at two different crossings, on the same side of that transversal. Same-side interior angles are also called co-interior angles or consecutive interior angles. Family names specify positions, even without parallelism.

Converse

A statement exchanging a conditional’s hypothesis and conclusion. A forward theorem needs a separately justified converse before it can be reversed.

Perpendicular distance and foot

Point-to-line distance is the perpendicular segment’s length; its endpoint on the line is the foot. A point on the line has distance zero. Between parallels, use a specified joining segment perpendicular to both.

Triangle side classifications

A triangle has three noncollinear vertices. Scalene means no congruent sides; isosceles means at least two congruent sides; equilateral means three congruent sides. Every equilateral triangle is isosceles.

Triangle angle classifications

Acute means three acute interior angles; right means one right interior angle; obtuse means one obtuse interior angle.

Auxiliary line and angle chase

An auxiliary line is added to reveal a relationship. An angle chase reaches unknown measures through a sequence of justified angle relationships.

Exterior and remote interior angles

A triangle exterior angle lies between one side and an adjacent side’s continuation beyond their common vertex. Its remote interior angles are at the other two vertices; the interior angle at its own vertex is adjacent.

Formulas and theorems

• Complement and supplement measures

  $$\begin{gathered} \text{Complement: }90^\circ-a\\ \text{Supplement: }180^\circ-a \end{gathered}$$

  Use when For an angle of measure $a$. A positive complement requires $0^\circ\le a<90^\circ$; a positive nonstraight supplement requires $0^\circ<a<180^\circ$.

  e.g. Find the complement of $59^\circ$: $31^\circ$.

• Linear-pair sum

  $$a+b=180^\circ$$

  Use when The two angles must be adjacent with opposite noncommon rays. Conversely, adjacent supplementary angles have opposite outer rays and form a linear pair.

• Vertical-angle theorem

  $$m\angle AOB=m\angle COD$$

  Use when Two distinct lines intersect at $O$, with $A,O,C$ in that order on one and $B,O,D$ in that order on the other. This is proved, not inferred from appearance.

• Congruent complements and supplements

  Complements of congruent angles are congruent. Supplements of congruent angles are congruent.

  Use when Use the same partner type on both sides. This includes two complements, or two supplements, of the same angle.

• Accepted angle addition and complete turns

  Consecutive parts add to the selected whole. A complete turn totals $360^\circ$.

  Use when Count every sector exactly once, with no gap or overlapping interiors. A partial collection of angles around a point need not total a full turn.

• Accepted Euclidean parallel foundations

  Through an external point in a line’s plane, exactly one parallel to that line exists. If same-side interior angles total less than $180^\circ$, the crossed lines meet on that side when extended.

  Use when The angle form requires distinct coplanar lines crossed by a transversal at distinct points. These are accepted foundations; corresponding-angle congruence is a proved result.

• Forward parallel-line angle relationships

  For parallel lines, corresponding, alternate interior, and alternate exterior angles are congruent. Same-side interior and same-side exterior angles are supplementary.

  Text description

  Two distinct parallel lines, l above m, are crossed by a transversal t that descends from upper left to lower right. At the upper intersection, the northwest, northeast, southeast and southwest sectors are one, two, three and four. At the lower intersection, those positions are five, six, seven and eight. Corresponding pairs are one and five, two and six, three and seven, and four and eight. Alternate interior pairs are three and five, and four and six. Alternate exterior pairs are one and seven, and two and eight. Same-side interior pairs are four and five, and three and six. Same-side exterior pairs are one and eight, and two and seven.

  Use when Use two distinct parallel lines and one transversal crossing at distinct points. In the locator, upper northwest, northeast, southeast, southwest sectors are 1, 2, 3, 4; lower sectors in that order are 5, 6, 7, 8.

• Transversal converses

  One congruent corresponding or alternate pair proves parallelism. One supplementary same-side interior or exterior pair also proves parallelism.

  Use when The lines are distinct and coplanar, cut by the same transversal at distinct points. These are proved converses. A local vertical pair or linear pair does not test parallelism.

• Planar parallel and perpendicular relationships

  Distinct coplanar perpendiculars to the same line are parallel. The perpendicular transversal theorem states that a transversal perpendicular to one of two parallels is perpendicular to the other. Two distinct lines parallel to a third are parallel to each other.

  Use when All lines involved lie in one plane. For the third claim, the three lines are pairwise distinct. These are proved results. The two-perpendiculars claim can fail without coplanarity; all three proofs here use one plane.

• Triangle angle-sum theorem

  $$a+b+c=180^\circ$$

  Use when $a,b,c$ are the three interior measures of one nondegenerate Euclidean triangle; each is strictly between $0^\circ$ and $180^\circ$.

  e.g. Triangle angles are $53^\circ$, $82^\circ$, and $c$; find $c$: $45^\circ$.

• Third-angle theorem

  Two congruent angle pairs in two triangles force the remaining angle pair to be congruent.

  Use when Both triangles are nondegenerate Euclidean triangles. Match the given angle pairs first; this theorem supplies no side-length equality.

• Right and obtuse angle restrictions

  At most one interior angle of a triangle is right or obtuse. In a right triangle, its two other angles are acute and complementary.

  Use when The triangle is nondegenerate. Use the complementary total only when a right angle is given or established.

• Exterior angle theorem and inequality

  $$\begin{gathered} e=a+b\\ e>a\\ e>b\\ c+e=180^\circ \end{gathered}$$

  Text description

  Triangle ABC has A above the base from B to C. The base continues beyond C to D, so B, C and D lie on one line in that order. The remote interior angles have measures a at A and b at B, marked with blue solid arcs. The adjacent interior angle has measure c at C, marked with a plain arc. The non-reflex exterior opening from CA to CD has measure e, marked with a red dashed arc. The adjacent interior and exterior openings share CA; their other rays CB and CD are opposite.

  Use when In nondegenerate triangle $ABC$, extend $BC$ past $C$ to $D$, with $B,C,D$ in that order. The labels $a,b,c$ are interior measures at $A,B,C$; $e=m\angle ACD$ is non-reflex. The inequalities compare remote angles only.

  e.g. Remote angles are $27^\circ$ and $71^\circ$; find the exterior measure: $98^\circ$.

• Three triangle exterior angles and turning

  $$e_A+e_B+e_C=360^\circ$$

  Use when Choose one non-reflex exterior angle per vertex of one nondegenerate triangle. To interpret the measures as a turning record, follow its boundary once with all left turns or all right turns.

  e.g. A triangle’s three non-reflex exterior turns are all right turns. Two measure $124^\circ$ and $129^\circ$; find the third: $107^\circ$.

Problem types, step by step

Exam traps

• Trap Calling supplementary angles a linear pair automatically

  Fix A linear pair also needs adjacency and opposite noncommon rays. Shared vertices alone do not exclude overlapping interiors.

• Trap Equating a complement and a supplement of equal angles

  Fix Use two complements or two supplements for congruent partners. Mixing partner types uses different totals.

• Trap Using an angle family name as an equation

  Fix A family describes position. For forward equalities or sums across crossings, establish parallelism first.

• Trap Using a local fact to prove parallelism

  Fix Vertical angles and linear pairs hold at one crossing without parallelism. A converse must constrain both crossings.

• Trap Treating a measurement or the goal’s markings as a given

  Fix Use exact givens or proved facts. Approximate readings suggest a conjecture and do not prove exact congruence.

• Trap Dropping a plane condition or using an oblique distance

  Fix Keep coplanarity in the planar line theorems. Distance uses the specified perpendicular segment. Its shortest-length proof is deferred to 5.1; equal perpendicular separation is justified later with triangle congruence.

• Trap Using a triangle formula on three collinear points

  Fix Triangle vertices must be noncollinear. Reject zero or negative interior measures even when an algebraic equation balances.

• Trap Using side information as an unproved angle equality

  Fix Classify sides from stated side facts. Theorems connecting equal sides with equal opposite angles are proved in 4.3; third-angle congruence gives no side equality.

• Trap Adding the exterior angle and its remote angles to get a straight angle

  Fix The exterior measure equals the remote sum. Only the adjacent interior angle supplements that exterior angle.

• Trap Applying the exterior inequality to the adjacent interior angle

  Fix The exterior angle exceeds each remote interior angle separately. Its adjacent interior angle can be larger.

• Trap Checking turn measures while mixing directions

  Fix The other exterior choice at one vertex has the same positive measure but reverses that vertex’s turn direction. A boundary turning record needs consistent choices.