Question

which of the following triangles would require you to apply the law of cosines before applying the law of sines or other triangle properties in order to determine the missing information? select all that apply.
image of six triangles:
△abc: sides 7 mm, 10 mm, 13 mm
△def: angle 41°, side 9 m, side 6.35 in
△ghi: side 7 m, side 4 m, angle 52°
△jkl: side 21 mm, side 16 mm, angle 17°
△mno: side 26 cm, side 22 cm, angle 61°
△pqr: side 17 cm, angle 26°, angle 109°
options: □△abc
□△def
□△ghi
□△jkl
□△mno
□△pqr

Explanation:

Step1: Recall Law of Cosines/Sines Conditions

Law of Cosines is used when we know two sides and the included angle (SAS) or three sides (SSS) to find a missing side/angle. Law of Sines is used when we know a side - angle pair (AAS, ASA, SSA). We need to identify triangles where we can't directly use Law of Sines and need Law of Cosines first.

Step2: Analyze $\triangle ABC$

In $\triangle ABC$, we know sides $AB = 7\space mm$, $BC=10\space mm$, and $AC = 13\space mm$ (SSS). To find any angle, we need to use Law of Cosines first (since SSS is a case for Law of Cosines) before we can use Law of Sines or other properties.

Step3: Analyze $\triangle MNO$

In $\triangle MNO$, we know two sides $ON = 26\space cm$, $MN=22\space cm$ and the included angle $\angle O=61^{\circ}$ (SAS). To find the third side or other angles, we can use Law of Cosines first.

Step4: Analyze $\triangle JKL$

In $\triangle JKL$, we know two sides $JK = 16\space mm$, $KL = 21\space mm$ and the included angle $\angle K = 17^{\circ}$ (SAS). We need to use Law of Cosines first.

Step5: Analyze other triangles

• $\triangle GHI$: We know two sides and a non - included angle? Wait, no, $\triangle GHI$ has sides $IH = 7\space m$, $HG = 4\space m$ and angle $\angle H=52^{\circ}$ (SAS? Wait, $IH = 7$, $HG = 4$, angle at $H$ is $52^{\circ}$, so SAS. Wait, but maybe I misread. Wait, no, let's re - check. Wait, $\triangle DEF$: Wait, the labels are a bit confusing. Wait, the triangles with SSS or SAS (where we can't use Sines directly) are $\triangle ABC$ (SSS), $\triangle MNO$ (SAS), $\triangle JKL$ (SAS). Wait, the options are $\triangle ABC$, $\triangle DEF$, $\triangle GHI$, $\triangle JKL$, $\triangle MNO$, $\triangle PQR$. Wait, $\triangle PQR$: In $\triangle PQR$, we know two angles and a side? Wait, $\angle Q = 26^{\circ}$, $\angle P=109^{\circ}$, side $QR = 17\space cm$ (AAS), so we can use Law of Sines directly. $\triangle GHI$: If we have two sides and a non - included angle? Wait, no, the diagram for $\triangle GHI$: Let's assume $IH = 7$, $HG = 4$, angle at $H = 52^{\circ}$ (SAS), but maybe the side - angle relation is different. Wait, the key is that $\triangle ABC$ (SSS), $\triangle MNO$ (SAS), $\triangle JKL$ (SAS) need Law of Cosines first. But the options given are $\triangle ABC$, $\triangle DEF$, $\triangle GHI$, $\triangle JKL$, $\triangle MNO$, $\triangle PQR$. From the checkboxes, the triangles that need Law of Cosines first are $\triangle ABC$, $\triangle MNO$, $\triangle JKL$. But let's focus on the options. The triangles where we need to apply Law of Cosines before Sines or other properties are $\triangle ABC$ (SSS), $\triangle MNO$ (SAS), $\triangle JKL$ (SAS).

Answer:

$\triangle ABC$, $\triangle MNO$, $\triangle JKL$ (the checkboxes for $\triangle ABC$, $\triangle MNO$, $\triangle JKL$ should be selected)