Question

use the measures of the sides of each triangle to list the angles from smallest to largest.
3.) triangle pqr with vertices p, o, r; sides: pq=4, qr=8, pr=10
4.) triangle beg with vertices b, e, g; sides: be=9.8, eg=16.1, bg=17.5
5.) triangle nlt with vertices n, l, t; sides: nl=x, lt=2x, nt=2x+5
use the measures of the angles of each triangle to list the sides from smallest to largest.
6.) triangle pzw with vertices p, z, w; angles: ∠z=42°, ∠p=57°, ∠w=81°
7.) triangle xmt with vertices x, m, t; angles: ∠x=24°, ∠m=90°, ∠t=66°
8.) triangle adv with vertices a, d, v; angles: ∠d=(4x)°, ∠v=x°, ∠a=(4x+5)°

Explanation:

Problem 3:

Step1: Recall the triangle side - angle relationship (larger side opposite larger angle).

In $\triangle POR$, the side lengths are $OP = 4$, $OR = 8$, $PR = 10$.

Step2: Order the sides from smallest to largest.

The order of the sides from smallest to largest is $OP$, $OR$, $PR$.

Step3: Determine the opposite angles.

The angle opposite $OP$ is $\angle R$, the angle opposite $OR$ is $\angle P$, and the angle opposite $PR$ is $\angle O$.
Since larger side is opposite larger angle, the order of angles from smallest to largest is $\angle R$, $\angle P$, $\angle O$. Wait, no, wait. Wait, side $OP = 4$ (opposite $\angle R$), side $OR = 8$ (opposite $\angle P$), side $PR = 10$ (opposite $\angle O$). So as the sides increase: $4<8 < 10$, so the opposite angles increase: $\angle R<\angle P<\angle O$. But the labels of the angles: $\angle R$ is at $R$, $\angle P$ at $P$, $\angle O$ at $O$. Wait, maybe I mixed up. Let's re - label. In $\triangle PQR$ (wait, the triangle is $P - O - R$). So vertices are $P$, $O$, $R$. Sides: $PO = 4$, $OR = 8$, $PR = 10$. So angle opposite $PO$ (length 4) is $\angle R$, angle opposite $OR$ (length 8) is $\angle P$, angle opposite $PR$ (length 10) is $\angle O$. So since $4<8 < 10$, then $\angle R<\angle P<\angle O$. But the problem is to list the angles (the vertices' angles) from smallest to largest. So the angles are $\angle R$, $\angle P$, $\angle O$? Wait, no, maybe the original triangle is labeled as $P$, $O$, $R$ with $PO = 4$, $OR = 8$, $PR = 10$. So angle at $R$ is opposite $PO = 4$, angle at $P$ is opposite $OR = 8$, angle at $O$ is opposite $PR = 10$. So the order of angles from smallest to largest is $\angle R$, $\angle P$, $\angle O$. But the answer should be in terms of the angle labels. Wait, maybe the problem is to list the angles (the angles at each vertex) from smallest to largest. So the sides: $PO = 4$, $OR = 8$, $PR = 10$. So the angle opposite the shortest side ($PO = 4$) is the smallest angle, which is $\angle R$. The angle opposite the middle - length side ($OR = 8$) is $\angle P$. The angle opposite the longest side ($PR = 10$) is $\angle O$. So the order is $\angle R$, $\angle P$, $\angle O$. But maybe the labels are different. Wait, the triangle is $P - O - R$, with $P$ connected to $O$ (length 4), $O$ connected to $R$ (length 8), and $P$ connected to $R$ (length 10). So angle at $R$: between $OR$ and $PR$, angle at $P$: between $PO$ and $PR$, angle at $O$: between $PO$ and $OR$. So using the side - angle relationship (in a triangle, the larger the side, the larger the opposite angle), we have:

Side $PO = 4$ (opposite $\angle R$), side $OR = 8$ (opposite $\angle P$), side $PR = 10$ (opposite $\angle O$). So since $4<8 < 10$, then $\angle R<\angle P<\angle O$. So the angles from smallest to largest are $\angle R$, $\angle P$, $\angle O$. But the problem's blank is under the triangle, with $P$, $Q$ (wait, maybe a typo, the triangle is $P - O - R$, so maybe $Q$ is $O$). So the blanks are for the angles (maybe the angle labels). So if we consider the angles at $P$, $O$, $R$:

The side opposite $\angle P$ is $OR = 8$, opposite $\angle O$ is $PR = 10$, opposite $\angle R$ is $PO = 4$. So the order of angles (from smallest to largest) is $\angle R$, $\angle P$, $\angle O$. But the blanks are labeled $P$, $Q$ (maybe $O$), $R$. So if $Q$ is $O$, then the order is $\angle R$, $\angle P$, $\angle O$ (or $\angle R$, $\angle P$, $\angle Q$ if $Q = O$).

Problem 4:

Step1: Recall the triangle side - angle relationship (larger side opposite larger angle).

In $\triangle BEG$, the side lengths are $BE = 9.8$, $EG = 16.1$, $BG = 17.5$.

Step2: Order the sides from smallest to largest.

The order of the sides from smallest to largest is $BE$, $EG$, $BG$.

Step3: Determine the opposite angles.

The angle opposite $BE$ (length 9.8) is $\angle G$, the angle opposite $EG$ (length 16.1) is $\angle B$, the angle opposite $BG$ (length 17.5) is $\angle E$.
Since larger side is opposite larger angle, the order of angles from smallest to largest is $\angle G$, $\angle B$, $\angle E$. Wait, no: side $BE = 9.8$ (opposite $\angle G$), side $EG = 16.1$ (opposite $\angle B$), side $BG = 17.5$ (opposite $\angle E$). So as $9.8<16.1 < 17.5$, the opposite angles increase: $\angle G<\angle B<\angle E$. So the angles from smallest to largest are $\angle G$, $\angle B$, $\angle E$.

Problem 5:

Step1: Recall the triangle side - angle relationship (larger side opposite larger angle). First, we need to order the side lengths.

The side lengths of $\triangle NLT$ are $NL=x$, $LT = 2x$, $NT=2x + 5$.
Since $x>0$, we can compare the lengths: $x<2x<2x + 5$ (because for $x>0$, $2x-x=x>0$ and $(2x + 5)-2x = 5>0$).

Step2: Determine the opposite angles.

The angle opposite $NL=x$ is $\angle T$, the angle opposite $LT = 2x$ is $\angle N$, the angle opposite $NT=2x + 5$ is $\angle L$.
Since larger side is opposite larger angle, the order of angles from smallest to largest is $\angle T$, $\angle N$, $\angle L$.

Problem 6:

Answer:

Step1: First, find the value of $x$ using the angle - sum property of a triangle.

The sum of angles in a triangle is $180^{\circ}$. So $x+(4x)+(4x + 5)=180$.
Combine like terms: $9x+5 = 180$.
Subtract 5 from both sides: $9x=175$? Wait, no: $x + 4x+4x + 5=180\Rightarrow9x+5 = 180\Rightarrow9x=175\Rightarrow x=\frac{175}{9}\approx19.44^{\circ}$. Wait, no, maybe I made a mistake. Wait, $x+4x+(4x + 5)=180\Rightarrow9x + 5=180\Rightarrow9x=175\Rightarrow x=\frac{175}{9}\approx19.44$. Then the angles are: $\angle V=x\approx19.44^{\circ}$, $\angle D = 4x\approx77.78^{\circ}$, $\angle A=4x + 5\approx82.78^{\circ}$.

Step2: Order the angles from smallest to largest.

The order of the angles from smallest to largest is $\angle V$, $\angle D$, $\angle A$.

Step3: Determine the opposite sides.

The side opposite $\angle V$ is $AD$, the side opposite $\angle D$ is $AV$, the side opposite $\angle A$ is $DV$.
Since larger angle is opposite larger side, the order of sides from smallest to largest is $AD$, $AV$, $DV$.

Problem 3 Answer: $\angle R$, $\angle P$, $\angle O$ (or according to the label correction, if $Q = O$, then $\angle R$, $\angle P$, $\angle Q$)

Problem 4 Answer: $\angle G$, $\angle B$, $\angle E$

Problem 5 Answer: $\angle T$, $\angle N$, $\angle L$

Problem 6 Answer: $PW$, $ZW$, $PZ$

Problem 7 Answer: $MT$, $XM$, $XT$

Problem 8 Answer: $AD$, $AV$, $DV$