Geometric proofs

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Formula

Explanation

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Worked solution

1. Given triangle ABC with angle A = 50°, angle B = 60°, and side AB = 10 cm. The sum of angles in a triangle is 180°, so angle C = 180° - 50° - 60° = 70°.
2. Point D lies on the extension of AC beyond C, so angle ACB = 70° is adjacent to angle BCD. Since D is on the extension, angle BCD = 180° - 70° = 110°.
3. In triangle ABD, angle ABD = 70° (given). We now know angle at B in triangle ABD is 70°, and angle at A is part of angle BAC = 50°, but since D is on the extension, angle DAB is still 50° because it's the same angle at A.
4. Sum of angles in triangle ABD is 180°. So angle at D, angle ADB = 180° - angle DAB - angle ABD = 180° - 50° - 70° = 60°.
5. Now, in triangle ABD, we have: angle DAB = 50°, angle ABD = 70°, angle ADB = 60°. Since angle ABD = 70° and angle ADB = 60°, they are not equal, but we must check for isosceles property.
6. To prove triangle ABD is isosceles, we need two equal angles or two equal sides. We now examine side lengths using the Law of Sines: in any triangle, a/sin(A) = b/sin(B) = c/sin(C).
7. Apply Law of Sines in triangle ABD: AB / sin(angle ADB) = AD / sin(angle ABD). Substitute known values: AB = 10 cm, angle ADB = 60°, angle ABD = 70°.
8. So: 10 / sin(60°) = AD / sin(70°). Therefore, AD = 10 * sin(70°) / sin(60°).
9. Use exact values: sin(60°) = sqrt(3)/2, sin(70°) is not a standard angle, but we keep it symbolic for exactness. So AD = 10 * sin(70°) / (sqrt(3)/2) = 10 * sin(70°) * 2 / sqrt(3) = 20 * sin(70°) / sqrt(3).
10. Now, to prove triangle ABD is isosceles, check if any two angles are equal. We have angles: 50°, 70°, 60° — all different. But wait: reconsider the configuration.
11. Re-evaluate: in triangle ABC, angle C = 70°, and D is on extension of AC beyond C. So angle BCD = 110°, and in triangle BCD, we have angle at C = 110°, angle at B = 60° (from triangle ABC), but angle ABD = 70°, which is part of angle at B. Wait — angle at B in triangle ABC is 60°, and angle ABD = 70°, so point D must be such that angle between AB and BD is 70°, meaning angle CBD = angle ABD - angle ABC = 70° - 60° = 10°.
12. Now, in triangle BCD: angle at C = 110°, angle at B = 10°, so angle at D = 180° - 110° - 10° = 60°. So angle BDC = 60°. But in triangle ABD, angle at D is angle ADB = 60°, and now angle BDC = 60°, so angle ADB = angle BDC = 60°, and they are adjacent at D. But more importantly, in triangle ABD, we now see that angle at D is 60°, angle at A is 50°, angle at B is 70° — still not equal. But we must recheck the isosceles claim.

Reference answer: 10 * sin(70°) / sin(60°)

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