Download Relationships Within a Circle

William Low
Relationships Within a Circle
Theorem 1
Theorem 2
Theorem 3
The longest chord of a circle is a diameter.
Any perpendicular bisector of a chord of a circle is a diameter.
In a circle, any radius perpendicular to a chord divides that chord into two congruent
segments.
Theorem 4 In a circle, any radius perpendicular to a chord divides the subtended arc into two
congruent arcs.
Theorem 5 Two chords are congruent if they are equidistant from the centre of a circle.
Theorem 6 In a circle, congruent chords subtend congruent arcs and, conversely, congruent arcs are
subtended by congruent chords.
Theorem 7 In a circle, arcs located between two parallel chords are congruent.
Theorem 8 Any line tangent to a circle is perpendicular to the radius that shares the point of tangency.
Theorem 9 For any circle, two tangent segments originating from the same exterior point are
congruent.
Theorem 10 Two parallel lines, be they tangents or secants, intercept congruent arcs of a circle.
Relationships Involving Two Circles
Theorem 11 The circumferences of two circles have the same ratio as their radii.
Theorem 12 The areas of two circles have the same ratio as the squares of their radii.
Theorem 13 The measures of similar arcs of two circles have the same ratio as their radii.
Relationships Governing Angular Measures in a Circle
Theorem 14 In a circle, the measure of a central angle is equal to the measure of its intercepted arc.
Theorem 15 In a circle, the measure of an interior angle is equal to one-half the sum of the measures of
the arcs intercepted by the angle and its vertical angle.
Theorem 16 In a circle, the measure of an inscribed angle is equal to one-half the measure of its
intercepted arc.
Theorem 17 For any circle, the measure of an exterior angle is equal to one-half the difference of the
measures of its intercepted arcs.
Relationships Governing Measures in a Right Triangle
Theorem 18 The hypotenuse of a right triangle inscribed in a circle is always a diameter of that circle.
Theorem 19 In a right triangle, the length of the median to the hypotenuse, is one-half the length of the
hypotenuse.
Theorem 20 In a right triangle, the length of the side opposite a 30º angle is one-half the length of the
hypotenuse.
Theorem 21 A right triangle and the altitude to its hypotenuse form two right triangles that are similar to
the given triangle and to each other.
Theorem 22 The length of the altitude to the hypotenuse of a right triangle is the geometric mean
between the lengths of the segments of the hypotenuse.
Theorem 23 The length of a leg of a right triangle is the geometric mean between the measure of its
projection onto the hypotenuse and the measure of the hypotenuse.
Theorem 24 The product of the lengths of the legs of a right triangle is equal to the product of the length
of the hypotenuse and the length of the altitude to the hypotenuse.
C:\ My Documents\My Work\Adult Ed\Math Notes 536\Geometry Theorems
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William Low
Congruent Polygons
Theorem 25 If two lines are both perpendicular to a third line, then they are parallel to each other.
Theorem 26 Every point on the perpendicular bisector of a line segment is equidistant from the two end
points of the line segment.
Theorem 27 Every point on an angle bisector is equidistant from the sides of the angle.
Theorem 28 If the non-common sides of two adjacent angles form a straight line, the angles are
supplementary.
Theorem 29 Vertical angles are congruent.
Theorem 30 If a transversal intersects two parallel lines, then :
1. The alternate interior angles are congruent.
2. The alternate exterior angles are congruent.
3. The corresponding angles are congruent.
4. The interior angles located on the same side of the transversal are supplementary.
5. The exterior angles located on the same side of the transversal are supplementary.
Theorem 31 In an isosceles triangle, the angles opposite the congruent sides are congruent, and the sides
opposite the congruent angles are also congruent.
Theorem 32 In an isosceles triangle, the perpendicular bisector of the side adjacent to the congruent
angles is the median and is the altitude from the vertex opposite that side, as well as the
bisector of the angle opposite that side.
Theorem 33 The line segment joining the midpoints of two sides of a triangle is parallel to the third side
and its measure is half that of the third side.
Theorem 34 The sum of the measures of the interior angles of a triangle is 180º.
Theorem 35 The sum of the measures of the interior angles of a polygon is equal to the number of sides
minus two, multiplied by 180º.
Theorem 36 The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is
360º.
Theorem 37 The opposite (or non-consecutive) angles of a parallelogram are congruent.
Theorem 38 The opposite (or non-consecutive) sides of a parallelogram are congruent.
Theorem 39 The diagonals of a parallelogram bisect each other.
Theorem 40 The diagonals of a rectangle are congruent.
Theorem 41 The diagonals of a rhombus intersect at right angles.
Theorem 42 The diagonals of a square intersect at right angles and are congruent.
Similar Polygons
Theorem 43 Transversals intersected by parallel lines are divided into proportional segments.
Theorem 44 Any line parallel to one side of a triangle and passing through a point inside the triangle,
determines a second triangle similar to the first.
Theorem 45 The measures of all the corresponding segments in two similar triangles are proportional.
Theorem 46 Two polygons with congruent corresponding angles and proportional corresponding sides
are similar.
Theorem 47 The perimeters of two similar polygons have the same ratio as the measures of the
corresponding diagonals and the measures of the corresponding sides.
Theorem 48 The perimeters of similar regular polygons have the same ratio as the radii of their
circumscribed circles and as their apothems.
Theorem 49 The areas of similar regular polygons have the same ratio as the squares of the radii of their
circumscribed circles and as the squares of their apothems.
Theorem 50 The areas of similar polygons have the same ratio as the square as their ratio of similitude
(i.e. the square of the ratio of the measures of any two corresponding segments).
C:\ My Documents\My Work\Adult Ed\Math Notes 536\Geometry Theorems
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