Angles and Sides കോണുകളും വശങ്ങളും
The big question of this section: if two triangles have the same three angles, what can we say about their sides? The answer turns out to be beautifully simple, and it is the heart of the whole chapter.
- If all the sides of two triangles are equal, the angles are also equal (SSS congruence, തുല്യത്രികോണങ്ങൾ).
- But equal angles do not force equal sides – a small and a large set square have the same angles.
- A line parallel to one side of a triangle cuts the other two sides in the same ratio (സമാന്തരവരകൾ).
1. The experiment: put the small triangle on the big one
Cut two triangles with the same angles (80°, 60°, 40°) but different sizes from cardboard. Put the small one on the big one so that the 80° corners coincide. Watch what happens to the bottom sides.
ഒരേ കോണുകളുള്ള രണ്ടു ത്രികോണങ്ങളുടെ തുല്യമായ കോണുകൾക്കെതിരെയുള്ള വശങ്ങൾ ജോടികളാക്കിയെടുത്താൽ, ചെറിയ നീളങ്ങളെല്ലാം വലിയ നീളങ്ങളുടെ ഒരേ ഭാഗമാണ്.
If two triangles have the same angles, pair up the sides opposite equal angles. Then every small side is the same fraction of its big partner (equivalently, every big side is the same multiple of its small partner):
pa = qb = rcZoom into a photo on your phone: every length in the picture grows by the same factor, and nothing looks distorted because all the angles stay the same. That is exactly what Key idea 1 says for triangles. The roof trusses of a house, a big and a small set square, the two ends of a torch beam – all are triangles with the same angles at different sizes.
2. Which side pairs with which? Opposite equal angles
The pairing rule matters. Side a is opposite the 80° angle in the big triangle, so its partner is p, the side opposite 80° in the small triangle. A quick check: in any triangle the smallest angle faces the shortest side and the largest angle faces the longest side. So the sides, listed in order of size, automatically pair up correctly.
Drag the orange dots. The largest angle (red) always faces the longest side (red); the smallest angle (blue) faces the shortest side (blue).
ഒരേ കോണുകളുള്ള ത്രികോണങ്ങളുടെ വശങ്ങൾ വലുപ്പക്രമത്തിൽ ഒരേ അംശബന്ധത്തിലാണ്.
Triangles with the same angles have their sides, taken in order of size, in the same ratio.
3. Scale factor തോത്
Instead of saying "each side is ⅔ of the other", we say the scale factor is ⅔. If a side of 6 cm becomes 4 cm:
- from big to small the scale factor is 46 = 23;
- from small to big the scale factor is 64 = 112.
Calling the scale factor k, Key idea 1 becomes: a = kp, b = kq, c = kr.
The blue triangle has sides 4, 6, 8 cm. The green one has the same angles. Move the slider: all three sides change together, and all three ratios stay equal to k.
ഒരേ കോണുകളുള്ള ത്രികോണങ്ങളിൽ, തുല്യമായ കോണുകൾക്കെതിരെയുള്ള വശങ്ങളുടെ നീളം മാറുന്നത് ഒരേ തോതിലാണ്.
In triangles with the same angles, the sides opposite equal angles all change by the same scale factor.
4. Worked example A: draw a ¾-size copy (page 102–103)
A triangle has base 6 cm with angles 50° and 70° at its ends. Draw a triangle whose sides are all ¾ of this one.
We only shrank one side and copied the angles. Key idea 2 promises that the other two sides then come out ¾ automatically. Angles control the shape; one side controls the size.
5. Worked example B: find the missing sides (pages 103–104)
In ΔABC, AB = 8 cm, AC = 6 cm, BC = 4 cm. In ΔPQR, PQ = 3 cm and ∠P = ∠C, ∠Q = ∠A, ∠R = ∠B. Find PR and QR.
The two ways to pair sides
| Method | How | Result |
|---|---|---|
| By angles | x° faces BC = 4 and PR; y° faces AC = 6 and PQ = 3; z° faces AB = 8 and QR | scale = 3 ÷ 6 = ½ ⇒ PR = 2 cm, QR = 4 cm |
| By size order | ABC: 4 < 6 < 8. PQR: PQ = 3 is the middle side (it faces the middle angle y°) | middle sides 6 → 3 ⇒ scale ½ ⇒ smallest 4 → 2, largest 8 → 4 |
6. Special to triangles! ത്രികോണവിശേഷം
Is "same angles ⇒ sides in the same ratio" true for every shape? No. Try it with four-sided shapes:
All eight angles are 90°, yet the square's sides are in the ratio 1 : 1 and the rectangle's are not. A triangle is the only polygon whose angles fix its shape completely – that is why this chapter is about triangles.
7. Check yourself
- Same angles ⇒ the sides opposite equal angles are in the same ratio: p/a = q/b = r/c = 1/k.
- The ratio is the scale factor. Big → small uses 1/k, small → big uses k.
- Pair sides by the angles they face, or simply by size order (shortest ↔ shortest, longest ↔ longest).
- To copy a triangle at a new size, scale one side and copy the angles.
- This works only for triangles (a rectangle and a square have the same angles but different side ratios).
Next: Exercise 1 – all seven questions solved with animations ▶