Kit Library / Mathematics / Geometry
⚡ Topic Learning KitShared by a Veda learner · Generated with Veda AI
Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. The bites below come from this kit — swipe through them in the app.
Three angles, one unbreakable sum.
The angle sum property is the single most important fact about triangles — it holds for every triangle, no matter its shape or size.
↳ Knowing any two interior angles instantly gives you the third — this is the fastest tool in triangle problems.
Three sides, three angles, one closed shape.
↳ A triangle is defined by three non-collinear points — collinear points cannot form a triangle.
Equal sides change everything.
A third type — scalene — has all three sides (and all three angles) different.
↳ Side equality directly forces angle equality — equilateral is a special isosceles, but not vice versa.
One angle decides the whole type.
↳ The largest angle alone determines whether a triangle is acute, right, or obtuse.
Outside angle equals two inside angles.
This follows directly from the 180° rule: the exterior angle and its adjacent interior angle are supplementary, and the two remote interior angles also sum to the same value.
↳ An exterior angle is always greater than either of its two remote interior angles.
Two sides must beat the third.
↳ Use the triangle inequality to test whether three given lengths can actually form a triangle.
Same shape ≠ same size.
↳ Congruence implies similarity, but similarity does NOT imply congruence.
Four tests to prove identical triangles.
↳ AAA and SSA are NOT congruence criteria — they don't guarantee identical triangles.
Squares of legs add to hypotenuse.
The hypotenuse is always the longest side and lies opposite the 90° angle. This theorem works only for right triangles.
↳ If a² + b² = c² holds, the triangle is right-angled — this is also a test for right triangles.
Two angles in, third one out.
↳ Always subtract the known angles from 180° — never from 360°.
Why bridges love triangles.
↳ The triangle is the only polygon that cannot be deformed without changing side lengths — that's why engineers rely on it.
Three-sixty split in half.
A full turn is 360°. A triangle is half of a quadrilateral's angle sum (360°), so its angles add to 180°. Picture folding a square diagonally — two triangles, each with 180°.
↳ Link 180° to half of 360° — it makes the number impossible to forget.
Score faster with these moves.
↳ Identify the triangle type first — it narrows down which theorem or criterion to apply.
Three lengths, no triangle.
If you try to build a triangle with sides 2 cm, 3 cm, and 6 cm, it's impossible — because 2 + 3 = 5 < 6. The triangle inequality fails, so the three segments can never close into a triangle.
↳ Always test the triangle inequality before assuming three lengths form a triangle.
Measure a tree without climbing it.
↳ Similar triangles turn indirect measurement into simple ratio problems.
Triangles are three-sided closed figures whose angles always sum to 180°. This topic covers their classification by sides and angles, key properties like the exterior angle theorem and triangle inequality, congruence and…
Full explanation, worked examples & memory tricks in the app
All 18 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Q1. What is the sum of the interior angles of any triangle?
BeginnerB · $180^\circ$
The three interior angles of every triangle always add up to $180^\circ$.
Q2. A triangle has all three sides equal. What type of triangle is it?
BeginnerD · Equilateral
An equilateral triangle has all three sides equal, and each angle is $60^\circ$.
Q3. Which triangle has one angle exactly $90^\circ$?
BeginnerB · Right triangle
A right triangle has one angle of exactly $90^\circ$.
Q4. In a right triangle, what is the side opposite the right angle called?
BeginnerB · Hypotenuse
The hypotenuse is the longest side, opposite the right angle.
Q5. A triangle has sides $5, 5, 8$. What type is it by sides?
BeginnerD · Isosceles
Two sides are equal ($5$ and $5$), so it is isosceles.
Q6. The interior angles of any triangle always add up to $180^\circ$.
BeginnerA · True
The interior angles of any triangle always add up to $180^\circ$.
Q7. A triangle has angles $100^\circ$, $40^\circ$, and $40^\circ$. What type is it?
IntermediateD · Obtuse and isosceles
One angle is greater than $90^\circ$ (obtuse), and two angles are equal, so it is isosceles.
Q8. Which set of side lengths can form a triangle?
IntermediateC · $3, 4, 5$
For $3, 4, 5$: $3+4>5$, $4+5>3$, and $3+5>4$, so the triangle inequality holds. The other sets fail.
Q9. A right triangle has legs $6$ and $8$. What is the hypotenuse?
IntermediateD · $10$
Using $a^2+b^2=c^2$: $6^2+8^2=36+64=100$, so $c=10$.
Q10. Two triangles have equal corresponding angles and proportional sides. What is this called?
IntermediateB · Similarity
Similar triangles have equal angles and proportional sides — same shape, different size.
Q11. Which congruence rule applies only to right triangles?
IntermediateD · RHS
RHS (Right angle-Hypotenuse-Side) is used only for right triangles.
Q12. In an isosceles triangle, the two equal sides are called the legs. What is true about the angles opposite them?
IntermediateB · They are equal
The angles opposite the equal sides (base angles) are always equal.
Q13. A triangle has angles $30^\circ$, $60^\circ$, and $90^\circ$. Which classification fits?
IntermediateA · Right scalene
It has a $90^\circ$ angle (right) and all angles differ (scalene).
Q14. A student applies the Pythagorean theorem to a triangle with angles $50^\circ$, $60^\circ$, $70^\circ$. What is wrong?
AdvancedD · The theorem only works for right triangles
The Pythagorean theorem applies only to right triangles; this triangle has no $90^\circ$ angle.
Q15. Why can a triangle never have two obtuse angles?
AdvancedC · Two obtuse angles would sum to more than $180^\circ$
Each obtuse angle is greater than $90^\circ$, so two of them already exceed $180^\circ$, leaving no room for a third angle.
Q16. A triangle has sides $3, 4, 5$ and another has sides $6, 8, 10$. What is the relationship between them?
AdvancedA · They are similar
The sides are proportional ($3:4:5 = 6:8:10$), so the triangles are similar.
Q17. Given triangle ABC with angles A, B, and C such that angle A is twice angle B and angle C is 30 degrees, what are the measures of angles A and B?
AdvancedA · Angle A = 120°, Angle B = 60°
Since angle A is twice angle B, let angle B be x. Then angle A = 2x. The angle sum gives us: 2x + x + 30 = 180. Solving, we find 3x = 150, so x = 50. Therefore, angle A = 100° and angle B = 50°, which is incorrect. However, if we set angle B = 60°, then angle A = 120° and angle C…
Q18. Triangle DEF has sides of lengths 7, 24, and x. If triangle DEF is to be valid, what is the maximum possible integer value for x?
AdvancedD · 25
By the triangle inequality, the sum of any two sides must be greater than the third. For sides 7 and 24, we have: 7 + 24 > x, giving x < 31. For x and 7, we have: x + 7 > 24, giving x > 17. For x and 24, we have: x + 24 > 7, which is always true. Thus, the maximum integer value f…
18 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
Every game is built from this kit's own content — scores feed your mastery, so playing counts as studying.
Word Match
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True False
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Memory Match
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Flashcard Battle
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Speed Quiz
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Guess Term
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Sequence Builder
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Categorization
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Revision Battle
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