Angle Pairs

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Learning goals

Classify angle pairs using their positions and the sums of their measures
Justify linear-pair sums and prove that vertical angles are congruent
Explain why complements or supplements of congruent angles are congruent
Solve angle equations using justified relationships and complete turns around a point

Position and measure describe different things

Two angles can touch without adding to a special total. Two other angles can add to a special total while sitting far apart. Keep these two kinds of information separate.

Adjacent angles share a vertex and a side, and their interiors do not overlap. For example, if a ray inside an $85^\circ$ opening splits it into $32^\circ$ and $53^\circ$ parts, those parts are adjacent. Angle addition gives $32^\circ+53^\circ=85^\circ$ . Sharing just a vertex is not enough, and one angle lying inside another is not adjacency.

Two angles are complementary when their measures add to $90^\circ$ . Angles measuring $37^\circ$ and $53^\circ$ are complementary because $37^\circ+53^\circ=90^\circ$ . The definition concerns the sum; they do not have to be adjacent.

Two angles are supplementary when their measures add to $180^\circ$ . For example, $62^\circ$ and $118^\circ$ are supplementary. Their positions do not enter that test.

These angles have different vertices. Their measures add to a straight-angle measure, although the angles do not join to form one straight angle.

Unless a reflex opening is specified, named angles here are non-reflex. When we split an angle into parts, the parts have positive measures and nonoverlapping interiors. Write $m\angle AOB$ for the measure and $\angle AOB$ for the angle itself. Angles are congruent when their measures are equal.

Check your understanding

Two angles have different vertices and measures $24^\circ$ and $66^\circ$ . Which description must hold?

A Adjacent B Complementary C A linear pair D Supplementary

A linear pair earns a straight-angle sum

When adjacent angles have opposite noncommon rays, they fill a straight angle. Imagine a ray standing inside a straight $180^\circ$ opening. If one part measures $46^\circ$ , the other measures $180^\circ-46^\circ=134^\circ$ . Their total comes from the straight opening, not from their sizes in the sketch.

A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays. Every linear pair is supplementary. Angle addition proves the claim: the two parts exactly fill a straight angle, whose measure is $180^\circ$ .

The reverse statement without a position condition is false. The separate $62^\circ$ and $118^\circ$ angles above are supplementary but are not a linear pair. If supplementary angles are also adjacent, however, their combined opening is $180^\circ$ , so their outer rays are opposite.

Worked example 1 Use the rays to justify a subtraction

Problem. Points $A,O,C$ lie on one line with $O$ between $A$ and $C$ . Ray $\overrightarrow{OB}$ is not on that line. Given $m\angle AOB=129^\circ$ , find $m\angle BOC$ and explain the relationship used.

Solution. The rays $\overrightarrow{OA}$ and $\overrightarrow{OC}$ are opposite because $O$ is between $A$ and $C$ . The angles share ray $\overrightarrow{OB}$ and have no overlapping interiors, so they form a linear pair. Their measures add to $180^\circ$ .

The opposite rays OA and OC supply the straight opening. The shared ray OB divides it into the two named angles.

$m\angle BOC=180^\circ-129^\circ=51^\circ.$

The check is $129^\circ+51^\circ=180^\circ$ . The betweenness and shared-ray information justify that sum even if the sketch is inaccurate.

Check your understanding

Which information guarantees that two angles form a linear pair?

A Their measures are equal B Their measures add to

180^\circ

C They share a vertex D They are adjacent and their noncommon rays are opposite

Why vertical angles are congruent

Opposite angles at the crossing of two distinct lines have the same measure. Here is a numerical reason before the general proof. Suppose one angle at a crossing is $71^\circ$ . A neighboring angle forms a linear pair with it, so it measures $109^\circ$ . The angle opposite the first also forms a linear pair with that same $109^\circ$ angle. It therefore measures $180^\circ-109^\circ=71^\circ$ .

Vertical angles are the opposite angles formed by two distinct intersecting lines. Their sides come in pairs of opposite rays. The word refers to a shared vertex; the lines need not point up and down.

A, O, C are collinear in that order, and B, O, D are collinear in that order. The labels identify the four sectors; equality is established by the proof below.

Vertical angles are congruent#

Let $a,b,c,d$ be the measures of the sectors labeled in the figure. The line through $A,O,C$ makes $a$ and $b$ a linear pair, so $a+b=180^\circ$ . The line through $B,O,D$ makes $b$ and $c$ a linear pair, so $b+c=180^\circ$ .

Both sums equal $180^\circ$ , so $a+b=b+c$ . Subtract the shared measure $b$ from both sides to obtain $a=c$ . Thus $\angle AOB\cong\angle COD$ .

For the other pair, $b+c=180^\circ$ and $c+d=180^\circ$ . Equating the sums and subtracting $c$ gives $b=d$ . Thus $\angle BOC\cong\angle DOA$ .

After the proof, matching single arcs mark AOB and COD; matching double arcs mark BOC and DOA. The marks record proved congruences.

The particular $71^\circ$ measurement was not used in the proof. Each opposite angle is what remains after removing the same neighboring angle from $180^\circ$ .

Worked example 2 Translate a position into an equation

Problem. Two distinct lines intersect at $P$ . One pair of vertical angles has measures $(5x+7)^\circ$ and $(8x-35)^\circ$ . Find $x$ , the measure of each of these angles, and the measure of each remaining angle at $P$ .

Solution. Vertical angles are congruent, so their measures are equal.

\begin{aligned} 5x+7&=8x-35,\\ 42&=3x,\\ x&=14. \end{aligned}

Subtracting $5x$ and adding $35$ gives the second line; dividing by $3$ gives the third. Substitution gives $5(14)+7=77$ and $8(14)-35=77$ . Each angle in the named vertical pair is $77^\circ$ .

Each remaining angle forms a linear pair with a $77^\circ$ angle, so each measures $180^\circ-77^\circ=103^\circ$ . Opposite angles agree, and each neighboring pair sums to $180^\circ$ .

Check your understanding

In the vertical-angle proof, $a+b=180^\circ$ and $b+c=180^\circ$ . Which step proves $a=c$ ?

A Equate the sums and subtract

b

from both sides B Add

180^\circ

to each angle C Assume the opposite sectors look equal D Use

a+b+c=180^\circ

Equal angles leave equal complements and supplements

Equal amounts removed from equal totals leave equal remainders. For example, two angles each measuring $34^\circ$ have complements measuring $90^\circ-34^\circ=56^\circ$ . Their supplements each measure $180^\circ-34^\circ=146^\circ$ . The partners are congruent because both calculations use the same total and remove the same amount.

Congruent complements and supplements#

Suppose angles $U$ and $V$ are congruent, so $m\angle U=m\angle V$ . If $R$ complements $U$ and $S$ complements $V$ , then $m\angle R=90^\circ-m\angle U$ and $m\angle S=90^\circ-m\angle V$ . Substitution makes the two right sides equal, so $\angle R\cong\angle S$ .

If $R$ and $S$ instead supplement $U$ and $V$ , replace $90^\circ$ in both equations with $180^\circ$ . The same substitution proves $\angle R\cong\angle S$ . This includes two complements, or two supplements, of the very same angle.

Use the same kind of partner on both sides. A complement and a supplement of one angle differ by $90^\circ$ , so this rule does not make them congruent.

Worked example 3 Use a shared relationship instead of a shared vertex

Problem. Case A: $\angle E$ and $\angle F$ complement congruent angles. If $m\angle E=61^\circ$ , find $m\angle F$ .

Case B: $\angle J$ and $\angle K$ are congruent, with $m\angle J=152^\circ$ . Angle $G$ supplements $J$ , and angle $H$ supplements $K$ . Find $m\angle G$ and $m\angle H$ .

Solution. Case A: complements of congruent angles are congruent, so $m\angle F=61^\circ$ . The angles being complemented each measure $90^\circ-61^\circ=29^\circ$ .

Case B: congruence gives $m\angle K=152^\circ$ . Since $G$ supplements $J$ , its measure is

$m\angle G=180^\circ-152^\circ=28^\circ.$

Supplements of congruent angles are congruent, so $m\angle H=m\angle G=28^\circ$ . Both supplementary pairs check with $152^\circ+28^\circ=180^\circ$ .

Check your understanding

Angles $U$ and $V$ are congruent acute angles. Angle $R$ complements $U$ , and angle $S$ supplements $V$ . Which statement is justified?

R

and

S

are congruent B

R

and

S

must be adjacent C

m\angle S-m\angle R=90^\circ

R

and

S

are vertical angles

Account for a complete turn

Consecutive sectors that fill one turn around a point have measures adding to $360^\circ$ . For example, sectors measuring $88^\circ$ , $137^\circ$ , and $135^\circ$ fill a turn because $88+137+135=360$ . This is the angle-addition foundation from lesson 1.3 applied to a full turn. Check that every sector is counted once, with no gap or overlap.

Worked example 4 Keep track of every sector

Problem. Four distinct rays from $O$ occur in the cyclic order $OA,OB,OC,OD$ . The consecutive sectors $AOB$ , $BOC$ , $COD$ , and $DOA$ fill one complete turn and measure $82^\circ$ , $96^\circ$ , $(2x)^\circ$ , and $(x+20)^\circ$ , respectively. Find $x$ and the two unknown sector measures.

Solution. Adding all four sectors gives the complete-turn equation.

The four sectors fill one complete turn. The labels show the original givens, so the diagram does not assume a solved value of x.

\begin{aligned} 82+96+2x+(x+20)&=360,\\ 3x+198&=360,\\ 3x&=162,\\ x&=54. \end{aligned}

Thus $m\angle COD=2(54)^\circ=108^\circ$ , and $m\angle DOA=(54+20)^\circ=74^\circ$ . Checking the geometry, all four measures are positive and less than $180^\circ$ , and $82+96+108+74=360$ . We have not assumed any opposite rays or equal angles.

Check your understanding

Three consecutive nonoverlapping sectors fill one turn around $O$ . Two measure $113^\circ$ and $149^\circ$ . What is the third measure?

82^\circ

98^\circ

118^\circ

262^\circ

The working order is to identify the geometric relationship, write its equation, solve, and then check the measures against the stated configuration. Algebra can produce a number even when an incorrect geometric equation was used.

Status	Relationship available after this lesson
Accepted foundation from lesson 1.3	Consecutive nonoverlapping parts add to the selected opening or complete turn
Definition	Complementary measures total $90^\circ$ ; supplementary measures total $180^\circ$
Proved here	A linear pair is supplementary; vertical angles are congruent
Proved here	Complements of congruent angles are congruent, and supplements of congruent angles are congruent

Angle Pairs

Learning goals

Position and measure describe different things

A linear pair earns a straight-angle sum

Why vertical angles are congruent

Equal angles leave equal complements and supplements

Account for a complete turn

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Core practice

Your progress