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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Find the third angle.
- Match opposite side-angle pairs.
- Check side ordering against angle ordering.
Before you start
Triangle angle sum and evaluating sine in degree mode.
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 plane diagram has baseline AB = 10 m, angle A = 45°, and angle B = 75°. Find distances BC and AC to point C.
Why this math matters
Two viewing directions from a known baseline locate a point in a simple triangulation model. Matching each side to its opposite angle keeps the calculation organized.
Set up the model
A useful answer starts with clear assumptions:
- A, B, and C form a nondegenerate Euclidean triangle.
- Both angles are interior angles from the baseline toward C.
- The example assumes exact data and is not a field-survey accuracy claim.
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.
Locate a point from a baseline and two angles
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A plane diagram has baseline AB = 10 m, angle A = 45°, and angle B = 75°. Find distances BC and AC to point C.
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.
Complete the angles
C = 180° − 45° − 75° = 60°
The known baseline AB is opposite C, so its 10 m length pairs with sin60°, not with either baseline angle.
Find BC opposite A
BC = 10sin45°/sin60° ≈ 8.165 m
Set BC/sin45° equal to 10/sin60° and multiply by sin45°.
Find AC opposite B
AC = 10sin75°/sin60° ≈ 11.154 m
The largest angle is 75°, so AC should be the longest side. The ordering 8.165 < 10 < 11.154 agrees.
The result
BC is approximately 8.16 m and AC is approximately 11.15 m.
Two angles and one side determine this triangle's size and shape. With two sides and a nonincluded angle instead, a second triangle can sometimes fit the data.
Common mistakes to catch
- Pairing the baseline with angle A rather than opposite angle C produces a different equation.
- A negative or zero third angle means the proposed triangle is invalid.
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
A triangle has A = 30°, B = 60°, and a = 5. Find b.
Show a hint
Use b/sin60° = 5/sin30°.
Reveal answer and explanation
5√3, approximately 8.66
5(√3/2)/(1/2) = 5√3.
Practice 2
Can A = 30°, a = 3, and b = 10 form a triangle?
Show a hint
Compute sin B from the law of sines.
Reveal answer and explanation
No
sin B = 10sin30°/3 = 5/3, outside sine's range [−1, 1].
Take the idea with you
Draw labels before using a triangle formula, especially when a baseline is not named side a.
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: Find the gap between two angled paths
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