Angle Pairs: Practice

Complementary measures sum to $90^\circ$, giving $3x+4+2x+11=90$.

$5x+15=90$

Subtract 15 and divide by 5 to obtain $x=15$. The measures are $49^\circ$ and $41^\circ$, which sum to $90^\circ$.

Supplementary measures sum to $180^\circ$. Combining like terms gives $9x+18=180$.

$9x=162$

Dividing by 9 gives $x=18$. Substitution gives $78^\circ$ and $102^\circ$, with the required total.

Vertical angles have equal measures, so $7x-9=4x+30$. Subtract $4x$ and add 9.

$3x=39$

Thus $x=13$. Both expressions give $82^\circ$.

Congruence gives equal measures. Since the two equal measures total $180^\circ$, each is half that total.

$180^\circ\div2=90^\circ$

Their measures alone do not determine their positions.

The angles are adjacent with opposite outer rays, so their measures sum to $180^\circ$.

$5x+10=180$

Subtracting 10 and dividing by 5 gives $x=34$. Therefore $m\angle BOC=(3(34)-7)^\circ=95^\circ$. The other angle is $85^\circ$, and the two total $180^\circ$.

The bisector makes $m\angle BOC=36^\circ$. Angle addition gives $m\angle AOC=72^\circ$.

Because $OA$ and $OD$ are opposite rays, $\angle AOC$ and $\angle COD$ form a linear pair.

$m\angle COD=180^\circ-72^\circ=108^\circ$

The complete-turn sum gives $95+110+x+2x+8=360$, or $3x+213=360$.

$3x=147$

Thus $x=49$, and the requested measure is $(2(49)+8)^\circ=106^\circ$. Checking, $95+110+49+106=360$.

Supplements of the same angle are congruent, so $4x-6=2x+28$.

$2x=34$

Dividing gives $x=17$. Both $A$ and $B$ then measure $62^\circ$, and $C$ measures $118^\circ$.

The measure sum is known even without position information.

$40^\circ+50^\circ=90^\circ$

So the angles are complementary. A common vertex alone does not establish a shared side or nonoverlapping interiors, so the other position claims do not follow.

Neighboring sectors form a linear pair, so $14x+12=180$. Subtract 12 and divide by 14 to get $x=12$.

The measures are $(6(12)+9)^\circ=81^\circ$ and $(8(12)+3)^\circ=99^\circ$. The smaller is $81^\circ$.

$14x+12=180$

There are $2+3+4=9$ equal ratio parts in the $360^\circ$ turn. Each part measures $360^\circ\div9=40^\circ$.

The smallest sector has two parts, so its measure is $2(40^\circ)=80^\circ$.

$2(360^\circ\div9)=80^\circ$

The relevant relationship is that $Q$ and $R$ are supplementary.

$m\angle R=180^\circ-57^\circ=123^\circ$

The complement relationship would give $m\angle P=33^\circ$; it does not change the total used for $R$.