Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 10 chapters and video availability01 · Read and understand
What you will learn
- Use sine symmetry to identify two valid triangle configurations.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Algebraic equations, ratios, angle and area facts, and basic coordinate geometry.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A triangle has A=30°, opposite side a=5, and side b=8 opposite angle B. How many triangles are possible?
Why this math matters
Check uniqueness before treating limited triangle measurements as a complete specification. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use Euclidean geometry and the angle, parallelism, similarity, or congruence conditions stated; a sketch alone does not establish them.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
See why side-side-angle can give two triangles
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A triangle has A=30°, opposite side a=5, and side b=8 opposite angle B. How many triangles are possible?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
sin B=b sin A/a=8(1/2)/5=0.8
The sine rule translates the given side-angle pair into a constraint.
Develop the calculation
B≈53.13° or 126.87°
Sine has equal values at supplementary angles.
Check and interpret
Two triangles: C≈96.87° or 23.13°
Both remaining angles are positive, so neither candidate is excluded.
The result
Two triangles: C≈96.87° or 23.13°
Both remaining angles are positive, so neither candidate is excluded.
Common mistakes to catch
- The inverse-sine principal output can miss a second geometric solution.
- Every candidate must leave a positive third angle.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
With the same A and a but b=12, can a triangle exist?
Show a hint
Compute 12(1/2)/5.
Reveal answer and explanation
No
It would require sin B=1.2, outside sine's range.
Practice 2
Why does one inverse-sine calculator output not settle the main problem?
Show a hint
The principal value is only one angle with that sine.
Reveal answer and explanation
Its supplement must also be checked
Both supplementary candidates fit the remaining angle sum here.
Take the idea with you
Check uniqueness before treating limited triangle measurements as a complete specification.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Use angle agreement to scale triangles
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