High School Geometry

Geometry is the study of the relationships among points, lines, angles, planes or surfaces, and solids or volumes. It is a mathematical system that uses logical reasoning to discover new properties. It is a science that emphasizes how a solution to a problem can be found.

In this course you will review principles of those relationships and apply them to problem solving in Geometry.

Written by Thanom Boone, Ed.D., Memphis City Schools, Memphis, Tennessee
Eric B. Schmiedicke, Student, University of Tennessee - Knoxville

  1.    Introduction to Geometry
    Geometry is a study of relationships among figures. It uses logical reasoning to solve problems. We study Geometry by examining and proving rules and properties of geometric figures. We use logical reasoning rather than direct measurement.
  2.    Perpendicular Lines and Planes
    Point, Line, and Plane are three undefined terms in Geometry.

    A point is represented by a dot (.) and labeled with a capital letter. A line is a set of points staying in the same direction. A line is represented with arrowheads on each end and labeled with two capital letters with arrowheads above them.

    A plane is a flat surface that extends in all directions with no end. A plane is labeled with a capital letter.
  3.    Real Numbers
    In Geometry we use real numbers to describe size and distance. The real numbers are made up of the rational and the irrational numbers. A rational number can be expressed in the form a/b, where a and b are integers such as 1, 2, or 3. An irrational number cannot be expressed as a rational number.
  4.    Properties of Real Numbers
    The following properties are true for any number of a, b, and c.

    • Commutative Property of Addition: a + b = b + a
    • Commutative Property of Multiplication: ab = ba
    • Associative Property of Addition: (a + b) + c = a + (b + c)
    • Associative Property of Multiplication: (ab)c = a(bc)
    • Identity Property of Addition: a + 0 = a
    • Identity Property of Multiplication: a x 1 = a
    • Inverse Property of Addition: a + (-a) = 0
    • Inverse Property of Multiplication: a x 1/a = 1
    • Multiplicative Property of Zero: a x 0 = 0
    • Distributive Property: a(b + c) = ab + ac
  5.    Properties of Real Numbers (cont.)
    The following properties are true for any value of a, b, and c.

    • Addition Property of Equality: If a = b, then a + c = b + c
    • Subtraction Property of Equality: If a = b, then a - c = b - c
    • Multiplication Property of Equality: If a = b, then ac = bc
    • Division Property of Equality: If a = b and c not equal to 0, then, a/c = b/c
    • Reflexive Property of Equality: a = a
    • Symmetric Property of Equality: If a = b, then b = a
    • Transitive Property of Equality: If a = b and b = c, then a = c
  6.    The Ruler Postulate
    Points of a line can be described by real numbers in such a way that:

    1. To every point of the line there corresponds exactly one real number.
    2. The distance between two points is the absolute value of the difference of the corresponding numbers.
  7.    Line Segment, Midpoint, Ray
    1. A Line Segment (or segment) is a set of points that contain two endpoints and all the points between them.
    2. Ray (AB) is the part of line AB which starts at point A and extends without ending through point B.
    3. Congruent segments are two or more line segments whose lengths are equal.
    4. The midpoint of a segment is the point that divides the segment into two equal segments.
    5. A bisector of a segment is a line, segment, ray, or plane that intersects the segment at its midpoint.
  8.    Line, Plane, and Space
    Collinear:
    All points are collinear if they are on the same straight line. Any two points are collinear.
    Coplanar:
    All points are coplanar if they are on the same plane. Any three noncollinear points are coplanar.
    Space:
    Space is a set of all points. Four noncoplanar points determine space.
  9.    The Intersection of Lines and Planes
    A theorem is a statement (or statements) that can be proved. The following theorems have been proved and can be proved again. Remember these statements and use them to solve problems that follow.

    1. A line and a point not on the line are coplanar.
    2. Two intersecting lines are coplanar.
    3. Two planes intersect each other in a line.
  10.    Convex Set and Separation
    1. A set is called CONVEX if, for any two points X and Y, the entire segment XY lies within the set.
    2. A set is called NONCONVEX if, for any two points X and Y, the entire segment XY does not lie within the set.
    3. An EDGE is a LINE that separates a plane into two half planes.
    4. A FACE is a PLANE that separates space into two half spaces.
  11.    Angles
    An angle consists of two rays with the same endpoint. The two rays are called sides, and the endpoint is called a vertex. To name an angle, we use three capital letters. We name a point on one side, followed by the vertex and a point on the other side. To measure an angle, we use a protractor. A protractor measures an angle in degrees. Look at the figure # 1 and find the measure of each angle.
  12.    Supplementary Angles and Linear Pair
    Supplementary Angles are two angles whose sum of their measures is 180.

    A Linear Pair is two adjacent angles whose sum of their measures is 180.

    Note:
    A linear pair is always supplementary but not all supplementary angles are a linear pair. Two supplementary angles are a linear pair when they are adjacent and the sum of their measure is 180.
  13.    Complementary Angles
    1. An acute angle is the angle whose measure is greater than 0 but less than 90.
    2. An obtuse angle is the angle whose measure is greater than 90 but less than 180.
    3. Complementary angles are two angles whose sum of their measures is 90.
    4. If two angles are complementary, then each angle is acute.
  14.    Congruent Angles
    1. Two or more angles are congruent if they have the same measure.
    2. All right angles are congruent. (Because the measure of a right angle is 90 degrees).
    3. Supplements of congruent angles are congruent.
    4. Complements of congruent angles are congruent.
    5. Vertical angles are congruent.
  15.    Median and Angle Bisector
    1. A median of a triangle is a segment joining a vertex and the midpoint of the side opposite the vertex.
    2. An angle bisector of a triangle is a segment that divides an angle into two equal adjacent angles and joins the vertex of the bisected angle and its opposite side.
  16.    Included Angle, Included Side
    1. An angle of a triangle is said to be included between two sides if the two sides of the triangle form that angle.
    2. A side of a triangle is said to be included between two angles if the two specified angles are on both endpoints of the side.
  17.    Congruent Triangles SAS Postulate
    Two triangles are said to be congruent if all pairs of their corresponding parts are congruent. In other words, if three pairs of corresponding sides and three pairs of corresponding angles of two triangles are congruent, then the two triangles are congruent. However, with some combinations of corresponding parts for some triangles one may prove that two triangles are congruent.

    SAS Postulate: Two triangles are congruent if two pairs of their corresponding sides are congruent and a pair of their included angles are congruent.
  18.    Congruent Triangles LL, ASA, and SSS Theorems
    LL Theorem:
    Given two right triangles, if the legs (sides that make up right angle) of one triangle are congruent to the legs of the other triangle, then the two triangles are congruent.
    ASA Theorem:
    If two angles and the included side of one triangle are congruent to the corresponding angles and the included side of another triangle, then the two triangles are congruent.
    SSS Theorem: I
    f three sides of one triangle are congruent to the corresponding sides of another triangle, then the two triangles are congruent.
  19.    Congruent Triangles SAA, and HL Theorems
    SAA:
    If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
    HL:
    If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the two triangles are congruent.
  20.    Inequalities in Triangles
    Inequalities in a Single Triangle:
    For any triangle, if their sides or angles are not equal, then the longer side is opposite the larger angle and the larger angle is opposite the longer side.
    Inequalities in Two Triangles:
    If two corresponding sides of two triangles are equal but the included angles are not equal, then the third side of the larger triangle is longer than the third side of the smaller triangle; and the included angle of the larger triangle is larger than the included angle of the smaller triangle.
  21.    Altitudes
    An altitude of a triangle is a segment from a vertex of the triangle perpendicular to the line containing the opposite side of the triangle.

    An altitude is different from a median because a median of a triangle is a segment from a vertex of the triangle to the midpoint of the opposite side of the triangle.
  22.    Perpendicular Lines and Planes
    Two lines are perpendicular if they intersect each other and make the right angle.

    A line and a plane are perpendicular if they intersect and the line is perpendicular to any line lying in the plane that passes through the point of intersection.

    The distance from an external point to a plane is the length of the perpendicular segment from the point to the plane. The perpendicular segment is the shortest segment from the point to the plane.
  23.    Parallel Lines and Alternate Interior Angles
    1. Two or more lines which do not intersect are either parallel or skew.
    2. Parallel lines do not intersect and are on the same plane (coplanar).
    3. Skew lines do not intersect and are not coplanar.
    4. A transversal is a line that intersects two or more coplanar lines in different points.
    5. Alternate interior angles are two non-adjacent interior angles on opposite sides of the transversal.
    6. If two parallel lines are intersected by a transversal, then alternate interior angles are congruent.
  24.    Parallel Lines and Corresponding Angles
    1. Corresponding angles are two angles in corresponding positions relative to the two lines.
    2. If two parallel lines are intersected by a transversal, then corresponding angles are congruent.
    3. If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
  25.    Parallel Lines and Same-Side Interior Angles
    1. A transversal may intersect two lines and form angles. Same-side interior angles are two angles between the two lines on the same side of the transversal.
    2. If two parallel lines are cut by a transversal, then interior angles on the same side of the transversal are supplementary.
    3. If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.
  26.    Multi-Parallel Lines
    If three or more parallel lines intersect a transversal into equal segments, then these parallel lines intersect any transversal into equal segments.
  27.    Parallels and Quadrilaterals
    1. A quadrilateral is any four-sided figure such as a square, a rectangle, a rhombus, or a trapezoid.
    2. Square, rectangle, rhombus, and parallelogram are grouped into parallelograms because each figure has two parallel sides. A trapezoid is different because it has only one pair of parallel sides.
    3. Opposite sides of a parallelogram are parallel and congruent. Its opposite angles are congruent. Its consecutive angles are supplementary and its diagonals bisect each other.
    4. Diagonals of a square are perpendicular and bisect each other; so do the diagonals of a rhombus.
  28.    Parallel Planes
    1. Two planes are parallel if they do not intersect.
    2. Two parallel planes are everywhere equidistant.
    3. The intersection of a plane and two parallel planes is two parallel lines.
    4. If two planes are perpendicular to the same line, then the planes are parallel.
  29.    Dihedral Angles and Perpendicular Planes
    1. A dihedral angle is a union of two half planes (or faces) and a line (or edge) between two half planes.
    2. A dihedral angle is named by any two points on each half plane and two points on the line between the half planes.
    3. A plane angle is an angle formed by a dihedral angle intersecting a plane perpendicular to the edge of the dihedral angle.
    4. The measure of a dihedral angle is the measure of its plane angle.
  30.    Projections
    1. The projection of an object into a plane is the picture of the object on the plane as if the object blocks a ray of light perpendicular to
    2. The projection of a point is a point.
    3. The projection of a line into a plane is a line, unless the line is perpendicular to the plane. If the line is perpendicular to the plane, then the projection of the line is a point.
  31.    Angles of a Triangle
    1. The sum of the measures of the angles of a triangle is 180.
    2. The acute angles of a right triangle are complementary.
  32.    Exterior and Interior Angles of a Triangles
    1. An exterior angle of a triangle is formed by an intersection of a side of the triangle and another side extended.
    2. The measure of an exterior angle of a triangle is equal to the sum of its remote interior angles of the triangle.
    3. An exterior angle of a triangle and its adjacent interior angle are supplementary.
  33.    An Isosceles Triangle
    1. An isosceles triangle is a triangle that has at least two congruent sides. The congruent sides are legs, the other side is a base.
    2. The angles opposite equal sides of an isosceles triangle are equal.
  34.    Area of Rectangles
    1. The area of a rectangle is the product of its length (L) and width (W) or A = LW.
    2. The area of a square is the square of its side or A = Sý
  35.    Area of Triangles
    The area of a triangle is equal to one half of the product of its base and height.

    A = 1/2 x BH
  36.    Area of Parallelograms
    A parallelogram is a quadrilateral whose opposite sides are parallel and congruent.

    1. The area of a parallelogram is the product of its base and the corresponding altitude.
    2. The altitude (or height) of a parallelogram is the distance between its base and the side parallel to the base.
  37.    Area of Trapezoids
    1. A trapezoid is a quadrilateral with only one pair of parallel sides (called bases).
    2. The altitude of a trapezoid is the distance between its bases.
    3. The area of a trapezoid is one half of the product of its altitude and the sum of its bases (parallel sides), or A = 1/2 x a x (b1 + b2)
  38.    The Pythagorean Theorem
    1. The Pythagorean Theorem demonstrates that in a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the length of each leg.
    2. In a right triangle if the length of each leg is known, then the length of the hypotenuse is equal to the square root of the sum of the squares of the length of each leg.
    3. In a right triangle if the length of one leg and the length of the hypotenuse are known, then the length of the other leg is equal to the square root of the difference of the square of the hypotenuse and the square of the known leg.
  39.    The 30-60 Right Triangle Theorem
    In a right triangle, if the measures of the acute angles are 30 and 60, then

    1. the length of the hypotenuse is twice as long as the length of the shorter leg;
    2. the length of the longer leg is square root 3 times the length of the shorter leg.
  40.    The 45-45 Right Triangle Theorem
    1. In a 45-45 right triangle, the hypotenuse is square root 2 (or 1.414) times as long as a leg of the triangle.
    2. The legs of a 45-45 right triangle are congruent.
  41.    Ratios and Proportions
    1. A ratio is a comparison of two quantities by division in order that one can see how many times one quantity occurs related to another. Before division, the two quantities must be in the same unit of measurement.
    2. A proportion is an equation of two or more ratios.
    3. Cross Multiplication. If a/b = c/d, then ad = bc
    4. If a/b = c/d, then b/a = d/c, or c/a = d/b
    5. If a/b = c/d, then (a + b)/b = (c + d)/d, or (a - b)/b = (c - d)/d
    6. If a/b = c/d = e/f =..., then (a + c + e +...)/(b + d + f +...) = a/b = c/d = e/f = ...
  42.    Geometric Mean
    1. If a, b, and c are positive numbers and a/b = b/c, then b is the geometric mean of a and c.
    2. If b is the geometric mean of a and c, then b = û(ac).
  43.    Similar Triangles
    Two polygons are similar if they meet the following requirements:

    1. Their corresponding angles are congruent.
    2. Their corresponding sides are proportional.
  44.    AAA and AA Similarities in Triangles
    1. If all corresponding angles of two triangles are congruent, then the triangles are similar. (AAA Similarity)
    2. If two pairs of corresponding angles of two triangles are congruent, then the triangles are similar. (AA Similarity)
    3. A line parallel to one side of a triangle intersects the other two sides into two similar triangles.
  45.    SSS and SAS Similarities in Triangles
    1. If all corresponding sides of two triangles are proportional, then the triangles are similar. (SSS Similarity)
    2. If two pairs of corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar. (SAS Similarity)
  46.    Special Similarities in Right Triangles
    1. The altitude to the hypotenuse of any right triangle separates the triangle into two right triangles that are similar both to each other and to the original triangle.
    2. The altitude is the geometric mean of the segments into which it separates the hypotenuse.
    3. Either leg of the original right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.
  47.    Area Ratios in Similar Triangles
    The ratio of the areas of two similar triangles is the square of the ratio of any pair of corresponding sides.
  48.    Coordinate Plane: Distance and Midpoint
    1. The distance between points.
    2. The coordinates of the midpoint .
  49.    Slope
    1. The slope of a line is a number that shows a comparison of changing of y-values and x-values between two points on the line, or m = (Y2 - Y1)/(X2 - X1)
    2. If the ends of the line point to Quadrants I and III, then the slope is a positive number. If the ends of the line point to Quadrants II and IV, then the slope is a negative number.
    3. Parallel lines have the same slope.
    4. Slopes of perpendicular lines are negatively reciprocal.
  50.    Slope and Equation of a Line
    1. Y - Y1 = m(X - X1) is the equation of the line that passes through point (X1,Y1) and has the slope = m. (Point-Slope form)
    2. y = mx + b is the equation for the line with slope m and y-intercept at b. (Slope-Intercept form)
    3. AX + BY + C = 0 is the standard form of a linear equation, where A and B are not both zero.
  51.    Circle and Sphere
    1. A circle is a set of all coplanar points equidistant from a given point (center).
    2. A sphere is a set of all points in the space from a given point (center).
    3. A great circle is formed by a plane cutting a sphere through its center.
    4. A radius is the distance from the center to the circle.
    5. A diameter is two straight radii.
    6. A chord is a line segment whose endpoints are on the circle.
    7. A secant is a line intersecting a circle at two points.
    8. A tangent is a line intersecting a circle at one point.
  52.    Tangents and Chords of a Circle
    1. Every tangent to a circle is perpendicular to the radius drawn to the point of tangency.
    2. If a segment passes through the center of a circle and bisects a chord, then the segment is perpendicular to the chord. (The segment that passes through the center of a circle and that is perpendicular to a chord bisects the chord.)
    3. In a circle, two equidistant chords are congruent. (If two chords of the same circle are congruent, then they are equidistant from the center.)
  53.    Angles and Arcs of a Circle
    1. An arc is a part of a circle. The arc that is half of a circle is a semicircle. A minor arc is less than a semicircle. A major arc is greater than a semicircle.
    2. The measure of a minor arc is the measure of its central angle (the angle at the center of the circle intercepted by the arc).
    3. The measure of a major arc = 360 - the measure of its opposite minor arc.
    4. The measure of a semicircle is 180.
  54.    Inscribed Angles of a Circle
    1. An inscribed angle in an arc is an angle whose vertex is on the circle and the two endpoints of the arc lie on the two sides of the angle.
    2. The measure of an inscribed angle is half the measure of its intercepted arc.
    3. An angle inscribed in a semicircle is a right angle.
    4. Angles inscribed in the same arc are congruent.
    5. The opposite angles of an inscribed quadrilateral are supplementary.
  55.    Tangents, Secants, and Angles
    1. If a tangent and a secant intersect each other on the circle, then the measure of the angle formed by the intersection is one half the measure of the intercepted arc.
    2. If the intersection of two tangents or two secants, or a tangent and a secant, is outside the circle, then the measure of the angle formed by the intersection is one half the difference of the measures of the intercepted arcs.
    3. If the intersection of two secants is inside the circle, then the measure of the angle is one half the sum of the measures of the intercepted arcs.
  56.    The Products of Lengths of Chords, Secants, and Tangents
    1. If two chords of a circle intersect, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other.
    2. If two secants intersect outside a circle, then the product of the lengths of one secant segment and its external secant segment equals the product of the lengths of the other secant segment and its external secant segment.
    3. If a tangent and secant intersect outside a circle, then the square of the length of the tangent equals the product of the lengths of the secant and its external secant segment.
  57.    Equations of Circles
    1. The standard form of the equation of a circle
    2. A circle with center at (a,b) and radius = r is the graph of the equation (X - a )² + (Y - b)² = r²
  58.    Exterior and Interior Angles of a Polygon
    1. The sum of the measures of the exterior angles of any convex polygon is 360. (An exterior angle of a polygon is formed by the intersection of one side and the adjacent side extended.)
    2. The sum of the measures of the interior angles of any convex polygon with n sides is (n - 2)180.
  59.    Area of a Polygon
    1. The area of a regular polygon is one half the product of the apothem and the perimeter. A = 1/2 ap
    2. The apothem of a regular polygon is the perpendicular distance from the center of the polygon to a side.
  60.    Circumference and Area of a Circle
    1. The circumference of a circle is two times the product of ã and the length of the radius of the circle. C = 2(pi)r
    2. The area of a circle is the product of ã and square of the length of radius of the circle. A = (pi)r²
    3. (pi) is the ratio of the lengths of circumference and the diameter of the same circle. The approximate value is 3.14159...
  61.    Length of an Arc of a Circle
    1. An intercepted arc has the same measure (m) as its central angle.
    2. The length of an arc (L) = 2(pi)r x m/360, where pi = 3.14159265..., r = radius of the circle, m = measure of the arc (in degrees).
  62.    Areas of Sectors and Segments of a Circle
    1. The area of a sector is half of the product of its radius and the length of its arc. 
         A = 1/2 x rL  (r & L of arc are given)
or A = m/360 x (pi)r²  (m of arc & r are given)
or A = (m/360) x area of the circle.
    2. A segment of a circle is the region bounded by a chord and an arc intercepted by the chord of the same circle. Area of segment = Area of sector - Area of triangle (bounded by chord and sides of central angle)
  63.    Prisms: Surface Area and Volume
    1. A prism is a space figure consisting of two congruent bases of a polygon and lateral faces of rectangles or parallelograms. If they are rectangles, the prism is a right prism. Otherwise, it is an oblique prism.
    2. The lateral area of a right prism equals the perimeter of the base times the height of the prism. LA = ph Total area = LA + 2 x base area
    3. The volume of a right prism equals the area of a base times the height of the prism. V = Bh.
  64.    Pyramids: Surface Area and Volume
    1. A regular pyramid is the pyramid whose
      1. base is a regular polygonal region,
      2. altitude, the perpendicular line from the vertex to the base, meets the base at the center,
      3. lateral faces are congruent isosceles triangular regions,
      4. slant height is the altitude of each lateral face.
    2. Lateral area = 1/2 x slant height x perimeter of the base LA = 1/2 x lp
    3. Total area = lateral area + base area, or TA = LA + BA
    4. Volume = 1/3 x base area x altitude V = 1/3 x BAh
  65.    Cylinders: Surface Area and Volume
    For a right circular cylinder,

    1. Lateral Area = circumference of base x altitude
      LA = 2(pi)rh
    2. Total Area = lateral area + 2 x base area
      TA = 2(pi)rh + 2(pi)r²
      or TA = 2(pi)r(h + r) 3
    3. Volume = base area x altitude
      V = (pi)r²h
  66.    Cones: Surface Area and Volume
    For a right circular cone,

    1. Lateral Area = 1/2 x circumference of base x slant height
      LA = (pi)rl
    2. Total Area = lateral area + base area
      = (pi)rl + (pi)r²
      or TA = (pi)r(l + r)
    3. Volume = 1/3 x base area x altitude
      V = 1/3 x (pi)r²h
  67.    Spheres: Surface Area and Volume
    1. The surface area of a sphere is four times the area of a great circle.

      A = 4(pi)r²

    2. The volume of a sphere of radius r is 
                          3
       V = 4/3(pi)r
  68.    Transformations
    1. A transformation of the plane is a mapping with two properties; (a) Every point of the plane has exactly one image. (b) Every point of the plane has exactly one preimage.
    2. Isometry is a transformation without distorting distances.
    3. Translation is a transformation that maps any two points P and Q.
  69.    Trigonometric Ratios
    Let C be a right angle of a triangle ABC, and let a, b, c be the lengths of the sides opposite angles A, B, and C respectively.

    1. sine of angle A (sin A) = a/c = (opposite side to angle A)/hypotenuse
    2. cosine of angle A (cos A) = b/c = (adjacent side to angle A)/hypotenuse
    3. tangent of angle A (tan A) = a/b = (opposite side)/(adjacent side)
  70.    Cumulative Review
    This chapter provides you with questions from different chapters. You may use them to check yourself to see if you are able to apply those concepts and principles you have learned. However, the questions cover only portions of Geometry. You must be very careful when drawing conclusions from this review.