To solve these problems, we use the
Triangle Inequality of Angles rule. This rule is very simple:
* The
largest angle is always opposite the
longest side.
* The
smallest angle is always opposite the
shortest side.
* The middle angle is opposite the middle-length side.
So, to order the angles or find the largest/smallest one, we just need to look at the lengths of the sides.
Here is the step-by-step solution for each problem:
1) Triangle MIU
* Sides: $MI = 23$, $IU = 27$, $MU = 10$.
* Order of sides (longest to shortest): $27 > 23 > 10$.
* Therefore, the angles opposite them are in the same order: Angle $M$ (opposite 27), Angle $U$ (opposite 23), Angle $I$ (opposite 10).
*
Order: $\angle M, \angle U, \angle I$
2) Triangle PQY
* Sides: $PQ = 17$, $PY = 21$, $QY = 12$.
* Order of sides (longest to shortest): $21 > 17 > 12$.
* Angles opposite: Angle $Q$ (opposite 21), Angle $Y$ (opposite 17), Angle $P$ (opposite 12).
*
Order: $\angle Q, \angle Y, \angle P$
3) Triangle IHA
* Sides: $HA = 12$, $IH = 32$, $AI = 22$.
* Order of sides (longest to shortest): $32 > 22 > 12$.
* Angles opposite: Angle $A$ (opposite 32), Angle $H$ (opposite 22), Angle $I$ (opposite 12).
*
Order: $\angle A, \angle H, \angle I$
4) Triangle VBN
* Sides: $BN = 15$, $VB = 22$, $NV = 21$.
* Order of sides (longest to shortest): $22 > 21 > 15$.
* Angles opposite: Angle $N$ (opposite 22), Angle $B$ (opposite 21), Angle $V$ (opposite 15).
*
Order: $\angle N, \angle B, \angle V$
5) Triangle ADC
* Sides: $AD = 18$, $AC = 30$, $DC = 15$.
* Longest side: $30$ (Side $AC$). Opposite angle is
$\angle D$.
* Smallest side: $15$ (Side $DC$). Opposite angle is
$\angle A$.
*
Largest: $\angle D$
*
Smallest: $\angle A$
6) Triangle VJS
* Sides: $VJ = 20$, $VS = 37$, $JS = 21$.
* Longest side: $37$ (Side $VS$). Opposite angle is
$\angle J$.
* Smallest side: $20$ (Side $VJ$). Opposite angle is
$\angle S$.
*
Largest: $\angle J$
*
Smallest: $\angle S$
7) Triangle OGI
* Sides: $OG = 21$, $IO = 14$, $GI = 10$.
* Longest side: $21$ (Side $OG$). Opposite angle is
$\angle I$.
* Smallest side: $10$ (Side $GI$). Opposite angle is
$\angle O$.
*
Largest: $\angle I$
*
Smallest: $\angle O$
8) Triangle QSP
* Sides: $SP = 8$, $QS = 15$, $PQ = 14$.
* Longest side: $15$ (Side $QS$). Opposite angle is
$\angle P$.
* Smallest side: $8$ (Side $SP$). Opposite angle is
$\angle Q$.
*
Largest: $\angle P$
*
Smallest: $\angle Q$
Final Answer:
1) $\angle M, \angle U, \angle I$
2) $\angle Q, \angle Y, \angle P$
3) $\angle A, \angle H, \angle I$
4) $\angle N, \angle B, \angle V$
5) Largest: $\angle D$, Smallest: $\angle A$
6) Largest: $\angle J$, Smallest: $\angle S$
7) Largest: $\angle I$, Smallest: $\angle O$
8) Largest: $\angle P$, Smallest: $\angle Q$
Parent Tip: Review the logic above to help your child master the concept of triangle worksheet geometry.
