Geometry
Olympiad Geometry
Geometry Warm-Up: Angles and Lines

Geometry Warm-Up: Angles and Lines

Theory

To solve these problems, students should be familiar with:

  • Angle relationships formed by parallel lines and a transversal (corresponding, alternate, and co-interior angles).
  • Properties of angle bisectors and altitudes in triangles.
  • Internal angle sum formulas for polygons.
  • Basic geometric constructions involving squares and right triangles.
  • Angle properties in star polygons and composite figures.

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Problems

  1. Two parallel lines are intersected by a third line. Find the angle between the bisectors of the same-side internal angles.
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  2. Find the sum of the internal angles of:
    a) a quadrilateral;
    b) a convex n-gon.
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  3. In triangle ABC, ∠A = α. Find the angle between:
    a) the bisectors of angles B and C;
    b) the altitudes from vertices B and C (i.e., BM and CH).
    In part b), consider the cases where α < 90° and α > 90° separately.
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  4. Perfect (equilateral) triangles BCK and DCL are constructed externally on sides BC and CD of square ABCD. Prove that triangle AKL is also a perfect triangle.
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  5. On the extensions of the hypotenuse AB of right triangle ABC beyond points A and B, points K and M are placed such that AK = AC and BM = BC. Find ∠MCK.
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  6. Find the sum of the five angles at the tips of an irregular pentagram (five-pointed star).

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