Question

in exercises 13-16, list the sides of the given triangle from shortest to longest. (see example 4.)
13.
14.
15.
16.

Explanation:

For each triangle, we use the rule: In a triangle, the length of a side is proportional to the measure of its opposite angle (longer side opposite larger angle, shorter side opposite smaller angle).

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Problem 13:

Step1: Identify opposite angles/sides

• $\angle A=62^\circ$ opposite $BC$
• $\angle B=67^\circ$ opposite $AC$
• $\angle C=51^\circ$ opposite $AB$

Step2: Order angles smallest to largest

$51^\circ < 62^\circ < 67^\circ$

Step3: Match to opposite sides

$\angle C < \angle A < \angle B \implies AB < BC < AC$

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Problem 14:

Step1: Calculate missing angle

Sum of angles in triangle: $180^\circ - 112^\circ - 32^\circ = 36^\circ$ (this is $\angle Y$)

Step2: Identify opposite angles/sides

• $32^\circ$ opposite $XZ$
• $36^\circ$ opposite $YZ$
• $112^\circ$ opposite $XY$

Step3: Order angles smallest to largest

$32^\circ < 36^\circ < 112^\circ$

Step4: Match to opposite sides

$32^\circ < 36^\circ < 112^\circ \implies XZ < YZ < XY$

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Problem 15:

Step1: Calculate missing angle

Sum of angles in triangle: $180^\circ - 127^\circ - 29^\circ = 24^\circ$ (this is $\angle M$)

Step2: Identify opposite angles/sides

• $24^\circ$ opposite $NP$
• $29^\circ$ opposite $MN$
• $127^\circ$ opposite $MP$

Step3: Order angles smallest to largest

$24^\circ < 29^\circ < 127^\circ$

Step4: Match to opposite sides

$24^\circ < 29^\circ < 127^\circ \implies NP < MN < MP$

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Problem 16:

Step1: Calculate missing angle

Right angle is $90^\circ$, so missing angle: $180^\circ - 90^\circ - 33^\circ = 57^\circ$ (this is $\angle D$)

Step2: Identify opposite angles/sides

• $33^\circ$ opposite $FD$
• $57^\circ$ opposite $FI$
• $90^\circ$ opposite $DI$

Step3: Order angles smallest to largest

$33^\circ < 57^\circ < 90^\circ$

Step4: Match to opposite sides

$33^\circ < 57^\circ < 90^\circ \implies FD < FI < DI$

Answer:

1. $AB < BC < AC$
2. $XZ < YZ < XY$
3. $NP < MN < MP$
4. $FD < FI < DI$