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 alternate interior angles to justify the Euclidean triangle sum.
- 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
Triangle ABC has angles A=47° and B=68°. Prove the angle-sum rule used to find C.
Why this math matters
Identify the geometric assumptions behind a rule before using it in an unfamiliar setting. 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.
Prove the triangle angle sum with a parallel line
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Triangle ABC has angles A=47° and B=68°. Prove the angle-sum rule used to find 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.
Represent the conditions
Draw a line through C parallel to AB
A construction introduces a useful relation without changing the triangle.
Develop the calculation
The copied A and B angles plus C form a straight angle
Alternate interior angles identify the two copied angles.
Check and interpret
C=180°−47°−68°=65°
The calculation follows a Euclidean parallel-line argument rather than measurement of the sketch.
The result
C=180°−47°−68°=65°
The calculation follows a Euclidean parallel-line argument rather than measurement of the sketch.
Common mistakes to catch
- A drawn appearance does not establish parallelism; the construction does.
- The angle-sum theorem here belongs to Euclidean geometry.
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
What is an exterior angle adjacent to C in the main triangle?
Show a hint
It supplements sixty-five degrees.
Reveal answer and explanation
115°
It equals 180−65, also 47+68 by the exterior-angle theorem.
Practice 2
Does this proof directly establish the same sum for triangles drawn along great circles on a sphere?
Show a hint
The parallel-line construction is Euclidean.
Reveal answer and explanation
No
Spherical geometry has different line and angle relationships, so its triangle sums need not be 180 degrees.
Take the idea with you
Identify the geometric assumptions behind a rule before using it in an unfamiliar setting.
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 an included angle in a congruence argument
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