Honors Geometry Final Study Guide Notes

Interior Angle: The formula to calculate the interior angle of a polygon is given by:
$(n - 2) \times \frac{180}{n}$
where $n$ is the number of sides.
Exterior Angle: The formula to calculate the exterior angle of a regular polygon is:
$\frac{360}{n}$
where $n$ is the number of sides.

Examples:

For a 7-sided polygon:
$(7 - 2) \times \frac{180}{7} = \frac{900}{7} \approx 128.6^{\circ}$
For a 5-sided polygon:
$\frac{360}{5} = 72^{\circ}$

Practice:

Concepts:
- Properties of parallelograms, rectangles, rhombi, kites, and trapezoids.
- Diagonals, midpoints, and congruence.
Examples:
- Diagonals of parallelograms bisect each other but are not always congruent.
- In trapezoids, the midsegment is the average of the bases.
Practice:
1. Determine if a quadrilateral is a parallelogram using angle or side information.
2. If one diagonal of a rectangle is $5j + 7$ and the other is $-3j + 49$ , solve for $j$ .
3. Kite with diagonals 55 and 48: Find the full length using right triangles.

Concepts:
- Use coordinates to calculate lengths and slopes.
- Determine properties using coordinates.
- Diagonals and symmetry.
Practice:
1. Use the distance formula to find diagonal length.
2. Given a square's diagonal, find its perimeter.
3. Solve for missing coordinates when a midpoint or distance is known.

Concepts:
- Trig Ratios: $\sin = \frac{\text{opp}}{\text{hyp}}$ , $\cos = \frac{\text{adj}}{\text{hyp}}$ , $\tan = \frac{\text{opp}}{\text{adj}}$
- Solve right triangles using trig and inverse trig.
Examples:
- If $AC = 13$ and $BC = 12$ , use the Pythagorean theorem to find $AB$ , then find all trig ratios.
- Balloon climbs at $4^{\circ}$ over 50 miles: $h = 50 \cdot \sin(4^{\circ}) \approx 3.5$
Practice:
1. Express sin, cos, and tan for angle $\theta$ in a triangle.
2. Use inverse trig to find angles from side lengths.
3. Word problem involving angle of elevation.

Concepts:
- Arc classifications and angle measures.
- Circumference $= 2\pi r$ and area $= \pi r^2$ .
- Tangent properties and chord relationships.
Examples:
- Tangent segments from a point are congruent.
- Opposite angles of cyclic quadrilaterals are supplementary.
Practice:
1. Find angle measures using intercepted arcs.
2. Solve for arc lengths and circle segment relationships.

Concepts:
- Area of triangle, trapezoid, kite, regular polygon.
- Sectors of circles, composite shapes.
Examples:
- Sector area $= \frac{\theta}{360} \cdot \pi r^2$
Practice:
1. Calculate the area of the shaded region inside a circle.
2. Composite shapes: subtract inner from outer.
3. Area of a regular polygon with given side and apothem.

Concepts:
- Surface area = lateral area + base(s)
- Lateral area of prisms, cylinders, cones, pyramids.
Practice:
1. Find slant height using the Pythagorean Theorem.
2. Apply $LA = \pi r l$ for cones, $LA = ph$ for prisms.

Concepts:
- Volume of prisms: $V = Bh$
- Volume of cones/cylinders: $V = \pi r^2 h$ , spheres: $V = \frac{4}{3} \pi r^3$
- Convert scaled measurements.
Practice:
1. Volume of a cone with $r = 6$ and $h = 12$ .
2. Scaled object: 1:9 scale with 4 in. height → real height?
3. Volume of a composite solid (cube + pyramid).

Concept:
- Break down prisms into rectangles and bases.
Practice:
1. Calculate $SA$ by adding the area of all faces.
2. Use correct units and round to the nearest tenth.