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Theorem. The sum of the two acute angles in a right-angled triangle is 90°.
Theorem. If one of the acute angles in a right-angled triangle is α, find the angles between the height dropped to the hypotenuse and the triangle’s legs.
Theorem. If the median of a triangle is equal to half the length of the side it is drawn to, then the triangle is right-angled.
Theorem. In a right-angled triangle, the median to the hypotenuse is equal to half the hypotenuse.
Problems
1. The segments AM and BH are the median and height, respectively, of an acute triangle ABC. It is known that AH = 1 and 2∠MAC = ∠MCA. Find the side BC.
2. A ladder stands with its bottom on the floor and its top leaning against a vertical wall. As the ladder slides down due to gravity, what path does a kitten sitting in the middle of the ladder follow?
3. Find the acute angles of a right-angled triangle if the median to the hypotenuse divides the right angle in a ratio of 1:2.
4. In triangle DEF, the median DM equals half the side EF. One of the angles formed by the intersection of side EF with the bisector DL is 55°. Find the angles of triangle DEF.
5. Point C lies on a circle with diameter AB. Find angle ∠ACB.
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