Read the facts marked on the diagram
A printed triangle may not be drawn to scale. Use the stated lengths, angle values and conventional markings. A small square establishes a right angle; matching tick marks identify equal lengths. Parallel-line arrows supply a relationship that may justify an angle theorem. An angle that merely looks square is insufficient.
Separate given facts from conclusions. If a question asks you to prove two sides equal, their appearance cannot serve as the proof. When submitting a photo to Lirno, include the instruction and all markings so the explanation uses the same facts you have.
Find a hypotenuse or a missing leg
The Pythagorean theorem applies to a right triangle: a² + b² = c², with c opposite the right angle. For legs 6 cm and 8 cm, c² = 36 + 64 = 100, so c = 10 cm. A geometric length uses the positive root. The result should exceed either leg.
If the hypotenuse is 17 cm and one leg is 8 cm, the other leg b satisfies b² = 17² − 8² = 289 − 64 = 225. Thus b = 15 cm. Adding squares would be the wrong setup because 17 cm is already the hypotenuse. Label the sides before substituting.
Use triangle angles and straight-line angles
The interior angles of a triangle in ordinary plane geometry total 180°. If two are 48° and 67°, the third is 180° − 48° − 67° = 65°. Two adjacent angles that form a straight line also total 180°. An angle of 112° on that line leaves 68° for its adjacent partner.
Do not choose a rule from the numbers alone. Identify the triangle or straight line in the figure. When parallel lines are given, state which corresponding or alternate angles you are using. A written reason connects the angle calculation with the conditions that make it valid.
Match sides before using similarity
Similar triangles have equal corresponding angles and proportional corresponding side lengths. Suppose AB corresponds to DE, with AB = 4 cm and DE = 10 cm. The scale factor from the first triangle to the second is 10/4 = 2.5. If BC = 6 cm corresponds to EF, then EF = 6 × 2.5 = 15 cm.
Keep the direction of the ratio consistent. To return from the larger triangle to the smaller, divide by 2.5. Lengths scale by the factor, while areas scale by its square: a factor of 2.5 in corresponding lengths gives an area factor of 6.25. Rotation does not change which labeled parts correspond.
Distinguish area from perimeter
For a rectangle 7 cm long and 3 cm wide, area is 7 × 3 = 21 cm². Perimeter is 2(7 + 3) = 20 cm. Area measures the region; perimeter measures its boundary. The two answers can be numerically close while describing different quantities.
For a triangle, area is one half of base times perpendicular height. A base of 10 cm and perpendicular height of 4 cm gives 20 cm². A sloping side cannot replace the height unless it is perpendicular to that chosen base. An external altitude can still give the correct height.
Write a proof as connected statements
A proof needs a reason for each conclusion. For an isosceles triangle with two equal sides given, the angles opposite those sides are equal. If the remaining angle is 40°, the equal angles share the remaining 140°, giving 70° each. The equal-side condition justifies the equal-angle claim.
Ask Lirno to find the first unsupported statement in your proof rather than replace the whole proof. Check the theorem conditions, the correspondence of labels and the units of the final quantity. A useful explanation says why a relationship applies before carrying out its arithmetic.
Check whether the information determines one answer
Two side lengths alone do not determine a triangle. Sides of 6 cm and 8 cm give a hypotenuse of 10 cm only when the angle between them is stated to be 90°. If a third side c is unspecified, the triangle inequality requires 2 < c < 14 cm; it does not pick a single length. Read the right-angle mark before treating the example as a Pythagoras problem.
A uniqueness check can also help with area. A triangle with base 10 cm needs its perpendicular height to determine its area. A sloping side of 6 cm alone cannot be substituted as that height. If the figure appears to supply an answer but the labels do not, explain which additional fact you need rather than estimating from the picture.
