Honors Geometry 2025 Semester 2 Comprehensive Study Guide

Unit 6: Quadrilaterals

• Polygon Interior Angle-Sum Theorem: The sum of the interior angles of a convex polygon with $n$ sides is given by the formula:
    - $S = (n - 2) imes 180$

• Polygon Exterior Angle-Sum Theorem: The sum of the exterior angles of any convex polygon, one at each vertex, is always:
    - $360^{\circ}$

• Regular Polygons and Specific Measurements:
    - For a polygon where an interior angle is $156^{\circ}$, the number of sides can be determined using the formula $\frac{(n-2) \times 180}{n} = 156$.
    - Examples listed include polygons with $9$ sides, $15$ sides, and $24$ sides.
    - Specific interior angle calculations: $140^{\circ}$, $60^{\circ}$.
    - Measurement calculations include values such as $x = 8$, $34$, $14$, and angles of $90^{\circ}$ and $135^{\circ}$.

• Properties of Parallelograms and Other Quadrilaterals:
    - Consecutive angles are supplementary (sum to $180^{\circ}$).
    - Opposite sides and opposite angles are congruent.
    - Example segments and angles from the study guide:
      - $CB = 34$
      - $TR = 14$
      - $CE = 23$
      - $m\angle DEF = 117^{\circ}$
      - $VW = 27$
      - $x = 8$

Unit 7: Proportions and Similarity

• Topic #1: Similar Triangles:
    - Two triangles are similar ($\sim$) if their corresponding angles are congruent and their corresponding side lengths are proportional.
    - Example 1: Given $\Delta DEF \sim \Delta GHF$.
      - Setup: $\frac{9x-13}{15} = \frac{90}{27}$
      - Solving for $x$ results in $x = 7$.
    - Example 2: Given $\Delta PQR \sim \Delta TSU$.
      - Setup: $\frac{7x+2}{14} = \frac{35}{x+8}$
      - Solving for $x$ results in $x = 4$.
    - Example 3: Given $\Delta JKL \sim \Delta NML$.
      - Using proportions: $\frac{4x-12}{24} = \frac{2x+9}{21}$
      - Solving for $x$ results in $x = 13$.
      - To find $JL$, substitute $x$: $JL = 35$.

• Similarity Postulates and Theorems:
    - AA~ (Angle-Angle): Two triangles are similar if two angles of one triangle are congruent to two angles of another triangle.
    - SSS~ (Side-Side-Side): Two triangles are similar if all corresponding sides are proportional.
    - SAS~ (Side-Angle-Side): Two triangles are similar if two pairs of corresponding sides are proportional and the included angles are congruent.
    - Case Examples:
      - $\Delta VZW \sim \Delta XZY$: Identified as "yes" via AA~.
      - $\Delta EFD \sim \Delta ACB$: Identified as "yes" via SSS~.
      - $\Delta MNL \sim \Delta PKL$: Identified as "yes" via AA~.
      - Non-similar examples: "No, sides not proportional" or "No, angles not congruent."

Unit 9: Special Right Triangles and Trigonometry

• Pythagorean Theorem:
    - Used for right triangles to find a missing side: $a^2 + b^2 = c^2$.

• Trigonometric Ratios (SOH CAH TOA):
    - $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
    - $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
    - $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

• Law of Sines:
    - Applied when dealing with non-right triangles: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.
    - Problem 33: Finding $BC$. Result: $BC \approx 17$.
    - Problem 34: Finding $m\angle C$. Result: $m\angle C \approx 54^{\circ}$.

• Law of Cosines:
    - Applied when three sides or two sides and the included angle are known: $c^2 = a^2 + b^2 - 2ab\cos(C)$.
    - Problem 35: Finding the length of side $AB$.

• Miscellaneous Right Triangle Calculations:
    - Values provided: $54.3$, $22.2$, $4.4$, $4.9$, $38.8\text{ ft}$, $50.9$, $742.5\text{ ft}$, $79.9\text{ ft}$.

Unit 12: Circles

• Equation of a Circle:
    - Standard form: $(x - h)^2 + (y - k)^2 = r^2$
    - Example: A circle with center $(9, 13)$ and radius $r = 6$ has the equation $(x - 9)^2 + (y - 13)^2 = 36$.

• Angle and Arc Relationships:
    - Internal Angles and Arcs:
      - $m\angle MNJ = 76^{\circ}$, while associated arcs include $114^{\circ}$ and $38^{\circ}$.
      - $m\text{FE} = 137^{\circ}$ based on arcs of $87^{\circ}$ and $68^{\circ}$.
      - Inscribed angles: $m\angle EFG = 19^{\circ}, 49^{\circ}, 52^{\circ}$.
      - $m\text{SW}$ involves values $173^{\circ}, 53^{\circ}, 67^{\circ}$.
      - $m\text{AC}$ values: $65^{\circ}, 153^{\circ}$.
      - External angle involving secants/tangents: $m\angle PQR = 118^{\circ}$.
  

• Algebraic Circle Problems:
    - Problem 13: Given $m\text{EB} = (4x + 1)^{\circ}$, $m\text{DF} = (12x + 11)^{\circ}$, and $m\angle DCF = (9x - 1)^{\circ}$.
      - Using the property that the angle formed by two secants is half the difference of the intercepted arcs: $\frac{(12x + 11) - (4x + 1)}{2} = 9x - 1$.
      - $\frac{8x + 10}{2} = 9x - 1$
      - $4x + 5 = 9x - 1$
      - $6 = 5x$ (Note: Transcript solution provides $x = 7$ and $m\angle DCF = 62^{\circ}$, suggesting specific diagram configurations or different arc relations).

Unit 10 & 11: Areas of Polygons, Surface Area, and Volume

• Circle Formulas:
    - Circumference ($C$): $C = 2\pi r$ or $C = \pi d$.
      - Example: $C = 18.85\text{ cm}$.
    - Area ($A$): $A = \pi r^2$.
      - Example Area values: $245.72\text{ m}^2$, $395.11\text{ in}^2$, $72\pi$, $826.19\text{ cm}^2$, $615.75$.

• Solid Geometry (Surface Area and Volume):

  - 1. Rectangular Prism:
    - Dimensions: $22.5\text{ in}, 24\text{ in}, 14\text{ in}, 26.5\text{ in}$.
    - Volume ($V$): $3780\text{ in}^3$
    - Surface Area ($SA$): $1827\text{ in}^2$

  - 2. Cylinder:
    - Dimensions: Height $= 22\text{ mm}$, Diameter/Radius derived from $16\text{ mm}$.
    - Volume ($V$): $1408\pi\text{ mm}^3$
    - Surface Area ($SA$): $\approx 4423.36\text{ mm}^2$ (Calculated as $1507.96\pi$? per transcript context).

  - 3. Square Pyramid:
    - Dimensions: $16\text{ yd} \times 16\text{ yd}$ base, $20.8\text{ yd}$ slant height.
    - Volume ($V$): $1638.4\text{ yd}^3$
    - Surface Area ($SA$): $921.6\text{ yd}^2$

  - 4. Cone:
    - Dimensions: Height $= 9\text{ ft}$, Radius $= 5.6\text{ ft}$.
    - Volume ($V$): $295.56\text{ ft}^3$
    - Surface Area ($SA$): $285.01\text{ ft}^2$

  - 5. Triangular Pyramid:
    - Dimensions: $17\text{ cm}, 17.2\text{ cm}, 10\text{ cm}$.
    - Volume ($V$): $245.37\text{ cm}^3$
    - Surface Area ($SA$): $301.3\text{ cm}^2$

  - 6. Sphere (Large):
    - Dimension: Radius $= 15\text{ in}$ (Diameter $= 30\text{ in}$).
    - Volume ($V$): $14137.17\text{ in}^3$
    - Surface Area ($SA$): $2827.43\text{ in}^2$

  - 7. Sphere (Small):
    - Dimension: Radius/Diameter context provided as $4.5\text{ km}$.
    - Volume ($V$): $190.85\text{ km}^3$
    - Surface Area ($SA$): $190.85\text{ km}^2$