Based on TEKS for Geometry and McGraw-Hill Education Geometry Volume 2

Purposes of the Course: Students will be able to distinguish Undefined Terms, Definitions, Postulates, Conjectures, and Theorems.

• Compare relationships of Euclidean and Spherical Geometries
• Make conjectures using constructions
  • Bisectors of angles and parallel lines
  • Congruent angles and segments
• Investigate patterns of relationships
  • Criteria for Triangle Congruence
  • Special segments of a triangle
  • Diagonals of quadrilaterals
  • Special segments and angles of circles
• Identify and determine validity of converses, inverses, and contrapositives of conditional statements, AND the connection between a biconditional statement and a true conditional statement with a true converse.
• Construct congruent segments and angles, segment bisectors, angle bisectors, perpendicular lines, perpendicular bisectors of segments, lines parallel to a given line through a point not on a given line.
• Make conjectures about relationships using constructions of
  • Congruent segments
  • Congruent angles
  • Angle bisectors
  • Perpendicular bisectors
• Verify Triangle Inequality Theorem and apply it for problem solving
• Verify that a conjecture is false using a counterexample

Formulas and Essential Information

Equilateral triangles have 3 equal sides, and 3 equal interior angles summing 360 degrees, or 120 degrees each
Isosceles triangles have 2 equal sides, and 2 equal interior angles summing 360 degrees
Scalene triangles have no equal sides or equal angles, but the interior angles still sum 360 degrees
A quadrilateral has four sides, and four interior angles summing 360 degrees
The sum of interior angles of any polygon (“2D shape”) can be calculated using the formula 180(n-2)

• A Pentagon (5 sided shape) has interior angles summing 540 degrees, or 180(5-2)
• A Hexagon (6 sided shape) has interior angles summing 720 degrees, or 180(6-2)
• An Octagon (8 sided shape) has interior angles summing 1080 degrees, or 180(8-2)

The radius of a circle is half of its diameter, the distance across a circle at its widest
The circumference of a circle is calculated using the formula “2-pi-r” or “pi-d” where the diameter, or double the radius, is multiplied by the value of pi (usually simplified to 3.14)
The area of a circle is calculated using the formula “pi-r-squared” where pi is multiplied by the radius to the second power
The area of a sector of a circle is calculated by the formula πr² multiplied by the degrees of the sector angle divided by 360.

• The volume of a sphere is calculated by the formula: 4/3 πr³
• The surface area of a sphere is calculated by the formula: 4πr²
• The volume of a cone with a circular base is calculated by the formula: 1/3hπr²
• The surface area of a cone with a circular base is calculated by the formula: πr² + πrl = πr(l+r)
• The volume of a cylinder with circular bases is calculated by the formula: πr²h
• The surface area of a cylinder with circular bases is calculated by the formula: 2πr(r + h) or 2πr² + 2πrh

Pythageorean Theorem, or “a-squared plus b-squared equals c-squared”, is part of trigonometry, and supports the Triangle Inequality Theorem (for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side).

Supplementary angles add up to 180-degrees on either side of a bisector along a line
Complementary angles add up to 90-degrees
Vertical angles are on opposite sides of a bisector across a vertex, and are congruent