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• Below you will find study notes on:
  • Systems of Measurement
  • Geometry
  • and Circle Theorems

Systems of Measurement

• The English system measures length, weight, and volume.
  • It is mainly used in the United States and a few other countries.
• The the Metric system is an internationally agreed decimal system of measurement.
  • It was created in France in 1799.

Conversions

• To convert 56 feet to yards:
  • Knowing that 1 yard = 3 feet
  • Multiply 56 feet by (1 yard / 3 feet)
  • 56/3 = 18.6 yards (approx.)

Quadrilaterals

• A quadrilateral is a two-dimensional figure with four sides and four angles.
  • Quadrilaterals should be closed shapes.
  • All the interior angles of a quadrilateral sum up to 360 degrees.

Parallelograms

• Parallelograms are Quadrilaterals with opposite sides that are equal and parallel.
  • The opposite sides of a parallelogram are parallel and congruent.
  • Opposite angles are congruent.
  • Consecutive angles are supplementary.
  • The diagonals bisect each other.
  • Diagonals divide a parallelogram into two congruent triangles.

Parallelogram Theorems

• Opposite sides of a parallelogram are congruent.
• Opposite angles of a parallelogram are congruent.
• Consecutive angles in a parallelogram are supplementary.

Diagonals

• Diagonals are segments that join non-consecutive vertices.

Diagonal Properties

• When the diagonals of a parallelogram intersect, they meet at the midpoint of each diagonal.
  • Diagonals are not congruent.
  • The diagonals of a parallelogram bisect each other and form two congruent triangles.

Theorem of Rectangles

• Parallelograms with congruent diagonals are rectangles.
  • Parallelograms with at least one right angle are rectangles.

Theorem of Rhombus

• The diagonals of a rhombus are perpendicular.
  • Each diagonal of a rhombus bisects the opposite angles.
  • If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Theorems of a Square

• The diagonals of a square bisect the vertex angle.

Midline Theorem

• The line segment in a triangle joining the midpoint of any two sides of the triangle is parallel to its third side and is also half of the length of the third side.

Trapezoids

• Trapezoids are quadrilaterals with exactly one pair of parallel sides.
  • Parallel sides are called bases.
  • Angles formed by a base and a leg are base angles.
  • Non-parallel sides are called legs.

Isosceles Trapezoids

• An isosceles trapezoid is a trapezoid with congruent legs.
  • Both pairs of base angles of an isosceles trapezoid are congruent.
    • The diagonals of an isosceles trapezoid are congruent.

Isosceles Trapezoid Theorems

• Base angles of an isosceles trapezoid are congruent.
• Diagonals of an isosceles trapezoid are congruent.
• If the base angles of a trapezoid are congruent, then the trapezoid is isosceles.
• If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.

Median of a Trapezoid

• The median of a trapezoid is the segment joining the midpoints of the legs.
  • The median of a trapezoid is parallel to the bases.
  • The length of the median is one-half the sum of the lengths of the bases.

Kites

• A kite is a quadrilateral with two distinct pairs of congruent adjacent sides.
• If a quadrilateral is a kite, then its diagonals are perpendicular.
• If exactly one diagonal of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite.

Kite Area

• The area of a kite is half the product of the lengths of its diagonals.
  • Area (A) = (d1 * d2) / 2
    • Here d1 and d2 are the diagonals.

Example

• The area of a kite with diagonals of 2.5 cm and 4.5 cm is calculated as:
  • Area = 1/2 * (2.5 cm) * (4.5 cm) = 5.625 cm².
• If a quadrilateral ABCD is a kite and angle m∠2 = 51 degrees, then m∠3 = 39 degrees because:
  • The sum of angles in a kite around point A = 180 degrees.
  • m∠3 + 51 + 90 = 180.
  • m∠3 = 180 - 51 - 90 = 39.

The Parts of a Circle

• A circle is the set of all points in a plane that is the same distance from a given point, called the center of a circle.
  • This distance is called the radius of the circle.
  • A circle is named by its center.

Key Lengths

• The circumference is the distance around the outside of a circle.
• The diameter of a circle is the distance from one side to the other, passing through the center of the circle.
  • A diameter of a circle is twice the length of a radius, given by the formula d = 2r.

Circle Segments

• A chord is a segment touching the circumference of the circle at two points.
• The radius is the segment connecting the center of the circle and the circumference.
  • All radii of a circle are congruent.

Circle Lines

• A secant is a line that intersects a circle in exactly two points.
• A tangent is a straight line that touches the circle at exactly one point.
  • That point is the point of tangency.

Circle Parts

• A segment is the part of a circle between a chord and an arc.
• A sector is the part of a circle between two radii and an arc.
• An arc is the part of the circumference at the edge of a sector.
  • It consists of two points on the circle and all the points on the circle between those two points.

Measurement of Arcs

• A minor arc measures less than 180 degrees.
  • It's named with 2 points.
• A major arc measures greater than 180 degrees.
  • It's named with 3 points.
• A semicircle measures 180 degrees.
  • It's named with 3 points.

Arc Addition Postulate

• The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
  • mABC = mAB + mBC

Central Angle & Intercepted Arcs

• The measure of a central angle is EQUAL to the measure of the INTERCEPTED ARC.
  • A central angle is an angle whose vertex is the CENTER of the circle

Inscribed Angles

• They are angles whose vertex is on a circle and whose sides contain chords.
• The measure of an inscribed angle is half the measure of the intercepted arc.
• An angle formed by a chord and a tangent can be considered an inscribed angle.
• An angle inscribed in a semicircle is a right angle.

Congruent Arcs

• Congruent arcs are arcs of the same measure.

Circle Theorems

• If two chords of a circle or congruent circles are congruent, then their corresponding minor arcs are congruent.
• If two minor arcs of a circle or of congruent circles are congruent, then their corresponding chords are congruent.
• If two central angles of a circle or of congruent circles are congruent, then their corresponding arcs are congruent.
• If two minor arcs of a circle or of congruent circles are congruent, then their corresponding central angles are congruent.
• If two central angles of a circle or of congruent circles are congruent, then their corresponding chords are congruent.
• If two chords of a circle or of congruent circles are congruent, then their corresponding central angles are congruent.

Congurent Chords Theorem

• In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

Interior Intersection Theorem

• If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
  • Interior angle = 1/2 (arc 1 + arc 2)

Exterior Intersection Theorem

• If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. NOTE. The sum of the exterior angle and the arc near is equal to 180 degrees. This only applies if the exterior angle is made by two tangent lines.
  • Exterior angle = 1/2 (far arc – near arc)