Question

list the sides of each triangle in order from shortest to longest.
10.
11.
12.

1. $\triangle abc$, with

$m\angle a = 90$,
$m\angle b = 40$, and
$m\angle c = 50$

1. $\triangle def$, with

$m\angle d = 20$,
$m\angle e = 120$, and
$m\angle f = 40$

1. $\triangle xyz$, with

$m\angle x = 51$,
$m\angle y = 59$, and
$m\angle z = 70$

Answer:

To solve these problems, we use the Triangle Angle - Side Relationship: In a triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. We first find the measure of the third angle (if not given) and then order the angles from smallest to largest. Then, the sides opposite these angles will be in the same order (shortest to longest).

Problem 10: $\triangle MON$ (Angles: $\angle O = 45^\circ$, $\angle M = 75^\circ$, $\angle N = 180 - 45 - 75 = 60^\circ$)

• Step 1: Order angles from smallest to largest: $45^\circ$ ($\angle O$) $< 60^\circ$ ($\angle N$) $< 75^\circ$ ($\angle M$)
• Step 2: Sides opposite these angles: Side opposite $\angle O$ is $MN$, opposite $\angle N$ is $MO$, opposite $\angle M$ is $ON$
• Order of sides (shortest to longest): $MN < MO < ON$

Problem 11: $\triangle GFH$ (Angles: $\angle G = 28^\circ$, $\angle F = 110^\circ$, $\angle H = 180 - 28 - 110 = 42^\circ$)

• Step 1: Order angles from smallest to largest: $28^\circ$ ($\angle G$) $< 42^\circ$ ($\angle H$) $< 110^\circ$ ($\angle F$)
• Step 2: Sides opposite these angles: Side opposite $\angle G$ is $FH$, opposite $\angle H$ is $GF$, opposite $\angle F$ is $GH$
• Order of sides (shortest to longest): $FH < GF < GH$

Problem 12: $\triangle TUV$ (Right triangle, $\angle U = 90^\circ$, $\angle V = 30^\circ$, $\angle T = 180 - 90 - 30 = 60^\circ$)

• Step 1: Order angles from smallest to largest: $30^\circ$ ($\angle V$) $< 60^\circ$ ($\angle T$) $< 90^\circ$ ($\angle U$)
• Step 2: Sides opposite these angles: Side opposite $\angle V$ is $TU$, opposite $\angle T$ is $UV$, opposite $\angle U$ is $TV$
• Order of sides (shortest to longest): $TU < UV < TV$

Problem 13: $\triangle ABC$ (Angles: $\angle A = 90^\circ$, $\angle B = 40^\circ$, $\angle C = 50^\circ$)

• Step 1: Order angles from smallest to largest: $40^\circ$ ($\angle B$) $< 50^\circ$ ($\angle C$) $< 90^\circ$ ($\angle A$)
• Step 2: Sides opposite these angles: Side opposite $\angle B$ is $AC$, opposite $\angle C$ is $AB$, opposite $\angle A$ is $BC$
• Order of sides (shortest to longest): $AC < AB < BC$

Problem 14: $\triangle DEF$ (Angles: $\angle D = 20^\circ$, $\angle E = 120^\circ$, $\angle F = 40^\circ$)

• Step 1: Order angles from smallest to largest: $20^\circ$ ($\angle D$) $< 40^\circ$ ($\angle F$) $< 120^\circ$ ($\angle E$)
• Step 2: Sides opposite these angles: Side opposite $\angle D$ is $EF$, opposite $\angle F$ is $DE$, opposite $\angle E$ is $DF$
• Order of sides (shortest to longest): $EF < DE < DF$

Problem 13: $\triangle ABC$ (Angles: $\angle A = 90^\circ$, $\angle B = 40^\circ$, $\angle C = 50^\circ$)

• Step 1: Order angles from smallest to largest: $40^\circ$ ($\angle B$) $< 50^\circ$ ($\angle C$) $< 90^\circ$ ($\angle A$)
• Step 2: Sides opposite these angles: Side opposite $\angle B$ is $AC$, opposite $\angle C$ is $AB$, opposite $\angle A$ is $BC$
• Order of sides (shortest to longest): $AC < AB < BC$

Problem 14: $\triangle DEF$ (Angles: $\angle D = 20^\circ$, $\angle E = 120^\circ$, $\angle F = 40^\circ$)

• Step 1: Order angles from smallest to largest: $20^\circ$ ($\angle D$) $< 40^\circ$ ($\angle F$) $< 120^\circ$ ($\angle E$)
• Step 2: Sides opposite these angles: Side opposite $\angle D$ is $EF$, opposite $\angle F$ is $DE$, opposite $\angle E$ is $DF$
• Order of sides (shortest to longest): $EF < DE < DF$

Problem 15: $\triangle XYZ$ (Angles: $\angle X = 51^\circ$, $\angle Y = 59^\circ$, $\angle Z = 70^\circ$)

• Step 1: Order angles from smallest to largest: $51^\circ$ ($\angle X$) $< 59^\circ$ ($\angle Y$) $< 70^\circ$ ($\angle Z$)
• Step 2: Sides opposite these angles: Side opposite $\angle X$ is $YZ$, opposite $\angle Y$ is $XZ$, opposite $\angle Z$ is $XY$
• Order of sides (shortest to longest): $YZ < XZ < XY$

Final Answers (Summarized):

1. $\boldsymbol{MN < MO < ON}$
2. $\boldsymbol{FH < GF < GH}$
3. $\boldsymbol{TU < UV < TV}$
4. $\boldsymbol{AC < AB < BC}$
5. $\boldsymbol{EF < DE < DF}$
6. $\boldsymbol{YZ < XZ < XY}$