Download GEOMETRY - PROBLEMS I Angles (1) Find adjacent supplementary

GEOMETRY - PROBLEMS I
Angles
(1) Find adjacent supplementary angles for which the larger is twice the smaller.
(2) Find adjacent complementary angles for which the larger is 20◦ more than the smaller.
(3) Find two adjacent angles whose sum is 75◦ and whose difference is 21◦ .
(4) What is the angle between the fingers of a clock at 7 o’clock?
(5) Prove that vertical angles are congruent.
Triangles
(1) Prove that the interior angles of a triangle add to 180◦ .
(2) Prove that in an isosceles triangle two interior angles are congruent.
(3) Prove that an isosceles triangle with one 60◦ angle is equilateral.
(4) In a triangle ABC, suppose that M is the midpoint on BC and 2|AM | = |BC|. Prove that ∠A = 90◦ .
(5) If a point lies on the bisector of an angle, prove that it is equidistant from the sides of the angle. Also
state and prove the converse.
(6) Prove that the angle bisectors of any triangle intersect at a point I, equidistant from all the sides of the
triangle.
(7) If a point lies on the perpendicular bisector of a segment, prove that it is equidistant from the endpoints
of the segment. Also state and prove the converse.
(8) Prove that the perpendicular bisectors of any triangle intersect at a point O, equidistant from all the
vertices of the triangle.
(9) A midsegment of a triangle is a segment connecting the midpoints of two sides of the triangle. Prove
that the midsegment is parallel to, and half the size of the third side of the triangle.
(10) Prove that in any triangle, the intersection point of two medians divides each median in the ratio 2 : 1.
(11) Prove that the three medians of a triangle meet in one point.
Quadrilaterals
(1) Prove that the interior angles of a quadrilateral add to 360◦ .
(2) Prove that a diagonal of a parallelogram divides it into two congruent triangles.
(3) If two sides of a quadrilateral are equal and parallel to each other, prove that the quadrilateral is a
parallelogram.
(4) Prove that in a parallelogram the diagonals bisect each other. Also state and prove the converse.
(5) Prove that a parallelogram with congruent diagonals is a rectangle.
(6) Prove that a quadrilateral is a rhombus if and only if the diagonals bisect each other at a right angle.
(7) Suppose ABCD is a parallelogram, and E the midpoint of BC, and G the midpoint of AD. Suppose
further that F, H are points on BD such that EF ⊥ BD and GH ⊥ BD. Prove that then the triangles
BF E and GHD are congruent.
(8) Prove that the midsegment of a trapezium is half the sum of the parallel sides.
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GEOMETRY - PROBLEMS I
Circles
(1) Prove that if a radius bisects a chord, then it is perpendicular to the chord.
(2) Prove that a perpendicular bisector of a chord passes through the center of the circle.
(3) Prove that an inscribed angle is half its intercepted arc.
(4) Prove that the angle made by a chord with a tangent to the circle is half the arc situated between the
chord and the tangent.
(5) Prove that an angle inscribed in a semicircle is a right angle (Thales).
(6) Prove that in a cyclic quadrilateral opposite angles are supplementary.
(7) If two circles intersect at two points, prove that the line connecting their centres is the perpendicular
bisector of the common chord.