Unit 6: Geometry and Measurement Detailed Study Guide

Angle, Parallel Line, and Triangle Theories

Supplementary Angle Theorem (S.A.T)

• The sum of the angles of a straight line equals $180^\circ$.

• Formula/Equation: $\angle A + \angle B = 180^\circ$

Complementary Angle Theorem (C.A.T)

• The sum of the angles of a right angle equals $90^\circ$.

• Formula/Equation: $\angle A + \angle B = 90^\circ$

Opposite Angle Theorem (O.A.T)

• Opposite angles are equal.

• Equations: $\angle a = \angle b$ and $\angle c = \angle d$.

Parallel Line Theorems

• Note on Usage: When applying these patterns, one must be certain the lines are definitely parallel, not just appearing to be parallel.

• Transversal: A line that crosses or intersects two or more lines, each at a distinct point.

• Cointerior Angles (C-Pattern): The sum of the interior angles on the same side of the transversal is $180^\circ$.

  • Example: $\angle 2 + \angle 5 = 180^\circ$; $\angle 3 + \angle 8 = 180^\circ$

• Alternate Angles (Z-Pattern): Alternate angles are equal.

  • Example: $\angle 3 = \angle 5D$; $\angle 2 = \angle 8$

• Corresponding Angles (F-Pattern): Corresponding angles are equal.

  • Example: $\angle 4 = \angle 8$; $\angle 3 = \angle 7$; $\angle 1 = \angle 5$; $\angle 2 = \angle 6$

Triangle Classifications

• Scalene Triangles: Have no equal sides and no equal angles.

• Acute Triangles: Have three angles that are all less than $90^\circ$.

• Isosceles Triangles: Have two equal sides and two equal angles.

• Right Triangles: Have exactly one angle that is $90^\circ$.

• Equilateral Triangles: Have three equal sides and three equal angles.

• Obtuse Triangles: Have one angle between $90^\circ$ and $180^\circ$.

Triangle Theorems (A.S.T.T., I.T.T., E.A.S.T., E.A.T.)

• Terminology:

  • Vertex: Point where two or more sides meet.

  • Interior angle: Angle formed on the inside of a polygon by two sides meeting at a vertex.

  • Exterior angle: Angle formed on the outside of a geometric shape by extending one of the sides past a vertex.

• Angle Sum of a Triangle Theorem (A.S.T.T.): The sum of the interior angles in a triangle is $180^\circ$.

  • Formula: $\angle A + \angle B + \angle C = 180^\circ$

• Isosceles Triangle Theorem (I.T.T.): In an isosceles triangle, the angles opposite the equal sides are also equal.

  • Formula: $\angle a = \angle b$

• Exterior Angle Sum of a Triangle (E.A.S.T.): The sum of the exterior angles in a triangle is $360^\circ$.

  • Formula: $\angle a + \angle b + \angle c = 360^\circ$

• Exterior Angle of a Triangle (E.A.T.): Each exterior angle is equal to the sum of the remote interior angles (angles at the opposite vertices).

  • Formula: $x = \angle y + \angle z$

• ### Examples: Solving Unknown Angles

  • Example 1: Determining unknown angles and identifying theorems.

  • Example 2:

    • i) Write an equation to find $x$.

    • ii) Solve the equation.

    • iii) State the measures of unknown angles using values like $x - 3$, $3x - 3$, and $2x + 17$.

  • Example 3: Solve for $z$, $y$, and $x$ with angles given as $58^\circ$, $67^\circ$, $x + 5^\circ$, and $3x - 72^\circ$.

  • Example 5: Multiple Choice. What is the value of $x$? Options: a. $14^\circ$, b. $120^\circ$, c. $62^\circ$, d. $45^\circ$.

Quadrilaterals and the Pythagorean Theorem

Angle Relationships in Quadrilaterals

• Quadrilateral Definition: A polygon having four sides, four angles, and four vertices.

• The sum of the interior angles of a quadrilateral is $360^\circ$.

• The sum of the exterior angles of a quadrilateral is $360^\circ$.

• Example 1: Find measures for angles $a$, $b$, $c$, and $d$ using known values such as $72^\circ$, $61^\circ$, $97^\circ$, and $88^\circ$.

Triangle Properties and the Pythagorean Theorem

• Median: A line segment joining a vertex to the midpoint of the opposite side.

• Altitude: The perpendicular distance from a vertex to the opposite side.

• Right Bisector / Perpendicular Bisector: A line that divides another line segment into two equal parts at a $90^\circ$ angle.

• Pythagorean Theorem (Specific to Right Triangles):

  • Legs: Two adjacent sides that contain the $90^\circ$ angle.

  • Hypotenuse: The longest side of a right triangle, located opposite the $90^\circ$ angle.

  • Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides.

  • Equation: $c^2 = a^2 + b^2$

• ### Pythagorean Theorem Word Problems and Calculations

  • Example 1: Calculate $x$ for a triangle with sides $3$ and $4$, or $x$ for a triangle with sides $12$ and $5$.

  • Example 6 (Farmer Fred): Olaf cuts across a rectangular field measuring $1500\,m$ by $2\,km$ ($2000\,m$) from corner to corner. Determine how much shorter his trip was compared to walking around the perimeter.

  • Example 7: A right-angled triangle has a base of $1000\,cm$ ($10\,m$) and an area of $60\,m^2$. Determine the perimeter of the triangle rounded to 2 decimal places.

Properties and Angles in Circles

Terminology

• Circumference: The distance around a circle.

• Radius: A straight line from the center to the circumference.

• Diameter: A straight line passing from side to side through the center.

• Chord: A line segment connecting two points on the circumference.

• Secant: A straight line that cuts a curve in two or more parts.

• Tangent: A straight line that touches a curve at exactly one point; if extended, it does not cross the curve at that point.

• Vertex: A meeting point of two lines forming an angle.

Angles in a Circle

• Central Angle: Angle between 2 radii with the vertex at the center and endpoints on the circumference.

• Inscribed Angle: Angle formed by 2 chords with the vertex and 2 endpoints on the circle.

• Rule: An inscribed angle cannot be drawn through the origin of the circle.

• Theorem: When the endpoints are the same for both the central and inscribed angles: $\text{Central } \angle = 2 \times (\text{Inscribed } \angle)$.

• Inscribed Quadrilateral Theorem: A quadrilateral can be inscribed in a circle if and only if opposite angles are supplementary.

  • $\text{Inscribed } \angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$

• ### Comprehensive Circle Example (Example h)

  • Diameter $AE = 20\,cm$, Chord $DE = 16\,cm$, $AF = 7.2\,cm$, and $\angle BFE = 90^\circ$.

  • Determine: $\angle ADE$, length of $AD$, length of $BD$, and length of $DF$.

Area and Perimeter of 2D Shapes and Composite Figures

Definitions and Units

• Area: The number of square units needed to cover a surface. Final units must be squared (e.g., $cm^2$).

• Perimeter: The distance around a shape. Final units are to the exponent 1 (e.g., $cm$).

• ### Applications

  • Example 2: A square has a perimeter of $156\,m$. Find the area of the square ($s = \frac{156}{4}$, Area = $s^2$).

  • Example 3: A rectangle with length $14\,m$ and area $119\,m^2$ requires finding the width ($119 / 14$).

  • Example 4: A square has an area of $86.49\,cm^2$. Find the perimeter.

  • Example 5: George's backyard resodding. Calculate area for grass and perimeter for fencing.

  • Example 10: Emma paints a circular table with radius $1.5\,m$, featuring a central hole with radius $20\,cm$. Calculate required paint area ($\text{Area} = \Pi \times 1.5^2 - \Pi \times 0.2^2$).

Surface Area and Volume of 3D-Shapes

Cubes, Prisms, and Pyramids

• Units: Surface Area is expressed in $units^2$, while Volume is expressed in $units^3$.

• Cube (6 sides): $SA = 6b^2$; Volume = $b^3$.

• Prisms: 3D figure with two parallel congruent polygonal bases.

  • General Volume: $\text{Volume} = (\text{Area of base}) \times h$

  • Rectangular Prism:

    • $SA = 2lw + 2wh + 2lh$

    • $V = lwh$

  • Triangular Prism:

    • $SA = ah + bh + ch + bl$

    • $V = \frac{1}{2}blh$ or $\frac{blh}{2}$

Pyramids and Cones

• Pyramid Definition: A polyhedron with a polygonal base and triangular faces meeting at a common vertex.

• Volume (Any Pyramid): $V = \frac{1}{3} \times (\text{area of base}) \times h$

• Square-Based Pyramid:

  • $SA = 2bs + b^2$

  • $V = \frac{1}{3}b^2h$ (where $s$ is slant height).

• Cylinder:

  • $SA = 2\Pi r^2 + 2\Pi rh$

  • $V = \Pi r^2 h$

• Cone:

  • Curved lateral surface extending from circular base to vertex.

  • Height ($h$): Perpendicular distance from vertex to base.

  • Slant Height ($s$): Distance from vertex to edge of base.

  • Calculating Slant Height: $s^2 = h^2 + r^2$ (Pythagorean Theorem).

  • $SA = \Pi r^2 + \Pi rs$

  • $V = \frac{1}{3}\Pi r^2 h$

  • Note: Volume of a cone is $\frac{1}{3}$ the volume of a cylinder with the same radius and height.

• ### Specific Volume Examples

  • Example d: Conical pile of road salt, Volume = $1500\,m^3$, Radius = $15\,m$. Find Surface Area.

  • Example e: A barrel containing the salt has the same radius and height as the pile. Find the lateral surface area to be painted.