# Geometry Survival Guide

## Big Ideas

• Create triangles and be able to prove if a triangle is similar using proportions and similarity theorems, such as side-side-side (SSS) or side-angle-side (SAS). (Chapter 6)
• Be able to identify a quadrilateral and prove it is a parallelogram with different theorems. Know the properties of each different shape and use those to help you prove the shape. You will be expected to write conjectures as part of your proofs. (Chapter 7)
• Use the Pythagorean theorem and Trigonometric ratios to determine the length of sides of a triangle. (Chapter 8)
• Be able to identify parts of a circle and calculate the area and circumference of a circle. Determine arcs and angle within or around a circle. (Chapter 9)
• Identify different three dimensional objects, their cross sections, area and volumes. (Chapter 10)

## Things I Struggled with

1. The number one thing I struggled with was determining what shape a quadrilateral could be. I know that sounds simple, but it can get a little tricky when a square could be any shape, but not any shape could be a square. To remember this I had to study a diagram. (Chapter 7)
2. The second hardest thing for me was setting up a mathematical proof to prove the shape of the triangles. It was mostly setting them up in the right order. I practiced these a lot and asked my teacher for help when I got it wrong. (Chapter 6)
3. The last thing I struggled with was trigonometry functions. It was confusing for me when I had to use the inverse of the function. To help I used SOHCAHTOA and did a lot of practice problems. (Chapter 8)
4. It was hard for me to picture the cross sections of some prisms, pyramids, etc. I always drew a diagram to help me with this. (Chapter 10)
5. It was hard for me to get the hang of figuring out the radius of a circle by using the tangent lines on a circle. (Chapter 9)

## Tips!

1. Do all your homework! It really helps you get a feel for different problems you could encounter, and if you don't get one you will be able to ask for help before you see it on a test.
2. Find little ways to remember formulas. For remembering Trigonometry functions you can use SOH-CAH-TOA, which translates to: sine with opposite side and hypotenuse - cosine with adjacent side and hypotenuse - tangent with opposite and adjacent side.
3. Write down all of your steps for problems, so that if you mess up you can go back through and find your mistakes.
4. Draw lots of pictures! Pictures help so much to make it easier to solve problems. They especially help with triangles and trigonometry.

## Vocabulary

1. Inequalities: A mathematical sentence built from expressions using one or more of the symbols <, >, ≤, or ≥.
2. Dilation: A transformation that maintains the shape of a figure, but multiplies its dimensions by a chosen factor.
3. Similar polygons: Polygons with the same shape, but different size.
4. Midsegment of a triangle: Segment that connects two midpoints of a triangle.

## Theorems

1. If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.
2. If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
3. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is AC + BC >AB
4. Side- Side- Side (SSS) Similarity Theorem: If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.
5. Side- Angle- Side (SAS) Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
6. Postulate* Angle- Angle (AA) Similarity: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
7. Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
8. Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

## Formulas

1. Proportions: Use proportions to find the ratio of two triangles, which would reveal if they were similar.
2. When added together, the length of the two shortest sides of a triangle must equal more than the length of the largest side.

## Vocabulary

• Parallelogram: A quadrilateral with both pairs of opposite sides.
• Congruent: Coinciding at all points when superimposed.
• Supplementary Angles: Angles that add up to 180 degrees.
• Bisect: Split something in half.
• Quadrilateral: A shape with four sides.
• Rhombus: A parallelogram with four congruent sides.
• Rectangle: A parallelogram with four right angles.
• Square: A parallelogram with four congruent sides and congruent angles.
• Trapezoid: A quadrilateral with exactly one pair of parallel sides.
• Kite: A quadrilateral with two distinct pairs of congruent adjacent sides.

## Theorems

1. Interior Angles of a Quadrilateral: The sum of the measures of the interior angles of a quadrilateral is 360 degrees.
2. If a quadrilateral is a parallelogram, then its opposite sides are congruent.
3. If a quadrilateral is a parallelogram, then its opposite angles are congruent.
4. If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
5. If a quadrilateral is a parallelogram, then its diagonals bisect each other.
6. Proving quadrilaterals are parallelograms: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
7. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
8. If an angel of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
9. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
10. If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.
11. Proving a Rhombus is a Parallelogram: A parallelogram is a rhombus if and only if its diagonals are perpendicular.
12. A parallelogram is a rhombus if and only if each diagonal bisect a pair of opposite angles.
13. Proving a Rectangle is a Parallelogram: A parallelogram is a rectangle if and only if its diagonals are congruent.
14. Proving a Trapezoid is a Quadrilateral: If a trapezoid is isosceles, then each pair of base angles is congruent.
15. If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
16. A trapezoid is isosceles if and only if its diagonals are congruent.
17. Proving a Kite is a Quadrilateral: If a quadrilateral is a kite, then its diagonals are perpendicular.
18. If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

## Formulas

Areas
1. Square/ rectangle (parallelogram): A= bh
2. Trapezoid: A = 1/2h (b1+b2)
3. Rhombus/ Kite: A = 1/2d1+ d2
4. Triangle: A = 1/2 bh

## Properties

1. Properties of a Parallelogram: a quadrilateral with both pairs of opposite sides.
2. Properties of a Trapezoid: The parallel sides are the bases.
A trapezoid has two pairs of base angles.
The nonparallel sides of the trapezoid are the legs.

## Vocabulary

1. Trigonometry: Deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships.
2. Angle of Elevation: The angle above horizontal that an observer must look to see an object that is higher than the observer.
3. Angle of Depression: The angle below horizontal that an observer must look to see an object that is lower than the observer.
4. Radian: A measure of an angle in standard position whose terminal side intercepts an arc of a certain length.
5. Hypotenuse: The longest side of a triangle that is opposite the right angle.
6. Coterminal: Angle whose terminal sides coincide.
7. Reference Angle: An acute angle formed by the terminal side of the given angle and the x- axis.

## Theorems

1. In a 30-60-90 degree triangle the sides are in the ratio 1: 2: Square root of 3
2. Pythagorean Theorem: a^2 + b^2 = c^2

## Formulas

1. Pythagorean Theorem (Pythagorean triple): a(leg)^2 + b(leg)^2 = c(hypotenuse)^2. This is used to find the length of a missing side of a triangle.
2. Trigonometric Ratios:
SinX = Opposite leg/ Hypotenuse (csc = reciprocal of sin) CosX = Adjacent leg/ Hypotenuse (sec = reciprocal of cos) TanX = Opposite leg/ Adjacent leg (cot = reciprocal of tan)
3. Inverse Trigonometric Functions: Angle = Sin^-1 (Opp/ hyp) Angle = Cos^-1 (Adj/ hyp) Angle = Tan^-1 (Opp/ adj)
4. Converting Between Degrees and Radians:
To get from radians to degrees- multiply by pi radians/ 180 degrees
To get from degrees to radians- multiply by 180 degrees/ pi radians

## Vocabulary

1. Circle: Set of points all in a plane that are equidistant from a fixed point called the center.
2. Radius: A segment whose endpoints are the center of the circle and one point on the side.
3. Chord: A segment whose endpoints are two point on a circle.
4. Diameter: A chord that passes through the center of the circle.
5. Arc: A part of the circumference of a circle.
6. Semicircle: Half a circle.
7. Minor Arc: An arc that is less than 180 degrees.
8. Major Arc: An arc that is more than 180 degrees.
9. Central Angle: An angle with a vertex at the center of the circle.
10. Intercepted Arc: An arc where the angle crosses the circle.
11. Tangent: A line in the plane of a circle that intersects the circle in exactly one point.

## Theorems

1. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the poin of tangency.
2. Tangents from a common external point are congruent.
3. If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
4. If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
5. If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
6. A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
7. If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
8. If two chord intercept the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
9. If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

## Formulas

1. Circumference: (2pi)r or (pi)d
2. Area of Circle: (pi)r^2
3. Arc length: Arc = mArc/360 x 2(pi)r
4. Area if Sector: A = mArc/ 360 x (pi)r^2

## Properties

1. Measuring arcs: The measure of a minor arc is equal to the measure of its central angle.
2. Congruent arcs: Two equal arcs of the same circle or congruent circle are congruent.

## Vocabulary

1. Polyhedron: A solid that is bounded by polygons, called faces, that enclose a single region of space.
2. Prism: A poly with two congruent faces, called bases, that lie in parallel planes.
3. Right Prism: Shape where each lateral edge is perpendicular to the base.
4. Oblique Prism: Shape where each lateral edge is not perpendicular to the base.
5. Cross Section: A view into the inside of something made by cutting through it.
6. Apothem of the Polygon: The distance from the center to any side of the polygon.
7. Central Angle of a Regular Polygon: An angle who vertex is the center and whose sides contain two consecutive vertices of the polygon.

## Theorems

1. Volume of a Pyramid: The volume of a pyramid is V=1/3 Bh, where B is the area of the base and h is the height.
2. Volume of a Cone: The volume of a cone is V= 1/3Bh = 1/2(pi)r^2h, where B is the area of the base, h is the height, and r is the radius of the base.

## Formulas

Volumes
1. Prism: l x w x h = V
2. Pyramid: V = 1/3Bh
3. Cone: V = 1/3Bh = 1/2(pi)r^2h
4. Sphere: V = 4/3(pi)r^3
5. Hemisphere: V = 1/2 x 4/2 x (pi) x r^3

## Properties

1. Cavalieri's Principle: If two prisms have the property that all corresponding cross sections have the same area, then those prisms have the same volume.