Geometry Lesson

Beyond Right Triangles: Law of Sines, Law of Cosines & the Ambiguous Case

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High SchooludlCCSS.MATH.CONTENT.HSG.SRT.D.10, CCSS.MATH.CONTENT.HSG.SRT.D.1170 min

Learning Objective

Students will solve for missing sides and angles in oblique triangles by selecting and applying the Law of Sines or Law of Cosines, justify their choice of law, and analyze the ambiguous SSA case to determine the number of possible triangles — demonstrating understanding through at least one self-selected modality.

Lesson Overview

Students extend triangle trigonometry to oblique triangles using the Law of Sines and Law of Cosines, including reasoning through the ambiguous SSA case. Multiple entry points, visual supports, and choice-based expression ensure all learners access and demonstrate mastery.

Materials

Printed task packets (large-print version available)
Colored pencils or markers for diagram annotation
Scientific calculators (or Desmos scientific calculator on devices)
Reference card: Law of Sines / Law of Cosines formulas with labeled triangle diagram
Dry-erase markers and personal whiteboards (or sheet protectors over paper)
Rulers and protractors for physical triangle construction (ambiguous case)
Anchor chart: Decision flowchart — Which law do I use?
Optional: GeoGebra or Desmos geometry on devices for dynamic exploration
Sentence frames for ELL students: 'I chose ___ because I was given ___.'
Exit slip cards (half-sheet)

Scaffolded Task Progression

1Task 1

Below is an oblique triangle. Use the diagram to answer the questions. ```geometry {"figureType":"triangle","angles":{"A":48,"B":75},"vertexLabels":["A","B","C"],"angleLabels":{"A":"48°","B":"75°","C":"57°"},"sideLabels":{"a":"side a (opposite A)","b":"side b (opposite B)","c":"12"},"showAngles":true,"title":"Oblique Triangle ABC"} ``` 1. Which side is opposite angle $A$? Which side is opposite angle $B$? 2. Write the Law of Sines ratio that connects side $c$ and angle $C$ with side $a$ and angle $A$. 3. If $c = 12$ and $C = 57°$, $A = 48°$, set up (but do not solve) the equation to find side $a$.

2Task 2

**Guided Access — First Law of Sines Solve (AAS)** In triangle $PQR$: $P = 50°$, $Q = 64°$, and $p = 18$ cm (side opposite $P$). ```geometry {"figureType":"triangle","angles":{"P":50,"Q":64},"vertexLabels":["P","Q","R"],"angleLabels":{"P":"50°","Q":"64°"},"sideLabels":{"p":"18 cm"},"showAngles":true,"title":"Triangle PQR — AAS"} ``` Step-by-step scaffold: - **Step A:** Find angle $R$. - **Step B:** Set up $\frac{p}{\sin P} = \frac{q}{\sin Q}$ and solve for $q$. - **Step C:** Set up $\frac{p}{\sin P} = \frac{r}{\sin R}$ and solve for $r$. Round all answers to one decimal place.

3Task 3

Developing — Law of Sines: Finding an Angle (ASA context)

A surveyor needs to find angle

B

in triangle

A B C

where

a = 22

b = 30

m, and

A = 35°

.

Choose your entry point:
- Option A (supported): Use the formula strip on your reference card. Set up

\frac{s i n B}{b} = \frac{s i n A}{a}

, substitute, then apply

sin^{- 1}

.
- Option B (independent): Set up and solve entirely on your own.

Find angle

B

. Round to the nearest degree.

*Real-world hook: This triangle represents a property boundary. The surveyor knows two sides and one angle — can they find the missing angle?*

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