Geometry

Question 1

Lines and Vocab
* Line Segment
* Congruent Line segments

Answer 1

• Line Segment: Lines with an endpoint
• Congruent Line Segments: Line segements that are equal in length to each other

Question 2

Angles

Answer 2

• When two lines cross, the opposite angles = each other.
• Angles next to each other add up to 180°
• Supplementary Angles: Angles that add up to 180°
• Complementary Angles: Angles that add up to 90°
• Acute Angle: 0°<Angle<90°
• Obtuse Angle: 90°<Angle<180°

Question 3

Parallel Lines and Angles

Answer 3

• If we intersect two parallel lines with another line, the angles across = each other, and angles next to each other are supplementary

Question 4

Polygon

Answer 4

• A closed shape formed by 3 or more line segments
• The number of sides = the number of angles.
• A Regular Polygon is one in which all sides are equal and all interior angles are equal.

Examples:
* Triangle
* Square/Rectangle
* Pentagon
* Hexagon
* Heptagon
* Octagon
* Quadrilateral/Rhombus

Question 5

Triangle Degree Rules

Answer 5

• All interior angles add up to 180°
• Exterior Angle = Sum of two opposite interior angles

Question 6

Sum of Interior Angles

Answer 6

(n - 2) x 180°
n = number of sides

Question 7

Interior/Exterior Angle of Regular Polygon

Answer 7

Interior Angle: [(n-2) x 180°] ÷ n
Exterior Angle: 360° ÷ n

Exterior angles get smaller the more sides there are.

Question 8

Sum of Exterior Angles

Answer 8

Sum of Exterior Angles of any Polygon is 360°

Question 9

Angle and Side Lengths

Answer 9

• As the angle gets bigger, the opposite side does as well. As it gets smaller, the opposite side gets smaller too.
• The angle opposite the longest side will be the largest, and the angle opposite the shortest side will be the smallest.

Question 10

Triangle Inequality Theorem

Answer 10

The sum of any 2 sides > than the third side

Question 11

Congruent Triangles

Answer 11

Congruent triangles are triangles that are identical in both size and shape, meaning all their corresponding sides and angles are equal

• SSS: Side Side Side: When all 3 sides are the same
• SAS: Side - Angle in btwn - Side
• AAS: Angle - Angle - Side: 2 angles and 1 side are the same
• ASA: Angle - Side in btwn - Angle

Not Congruent:
* AAA: Angle - Angle - Angle
* ASS: Angle - Side - Side

Question 12

Pythagorean Triplets

Answer 12

Right triangles in which all three sides are integers.
Pythagorean Triplets are infinite

• 3 : 4: 5
• 5 : 12: 13
• 8 : 15: 17
• 7 : 24 : 25

Question 13

30-60-90 Triangles

Answer 13

Side opposite a 30° angle = x
Side opposite a 60° angle = x√3
Side opposite a 90° angle = 2x

Question 14

45-45-90 Triangles

Answer 14

Side opposite a 45° angle = x
Side opposite a 90° angle = x√2

Question 15

Equilateral Triangle Area

Answer 15

Area = ½ side x ½ side x √3

Question 16

Similar Triangles

Answer 16

• Two triangles that share the same three angles.
• Sides are proportional to each other (same ratio between their sides)

Question 17

Types of Quadrilaterals

Answer 17

• Squares/Rectangles
• Parallelogram: Opposite sides are = and parallel, Opposite angles are =, Sum of angles on the same side = 180°
• Rhombus: All sides are =. Opposite sides are parallel, Opposite angles are =, Sum of angles on the same side = 180°
• Trapezoid: One pair of parallel sides, Sum of angles on the same side = 180°

Question 18

Parallelogram Area

Answer 18

base x height (height needs to be perpendicular line)

Question 19

Trapezoid Area

Answer 19

Avg of 2 bases x height
(base 1 + base 2) ÷ 2 x height

Question 20

Regular Hexagon Area

Find the area of a regular hexagon with a side length of 8

Answer 20

A regular hexagon can be divided into six equilateral triangles, so just find the area of one equilateral triangle and x 6 to find the whole area.

(½ side x ½ side x √3) x 6

1. Find area of equilateral triangle: (8÷2) x (8÷2) x (√3) : 4 x 4 x √3 = 16√3
2. Multiply by 6 = 16√3 x 6 = 96√3

Question 21

Regular Polygon Area

Answer 21

A regular polygon can be divided into congruent triangles, so just find the area of one triangle and multiply by # of sides to find the whole area.

(base x height) ÷ 2 x # of sides

Question 22

Maximizing a Polygon’s Area

Answer 22

To Maximize a Certain Polygon’s Area, make it regular.
Regular polygon area > Non-regular polygon area

To Maximize the Area Regardless of Shape, just turn it into a circle.

Question 23

Circles Vocab

Answer 23

• Minor arc: Two points lying on the circumference is an arc. The shortest is the minor arc
• Major arc: The longer one is the major arc.
• Chord: A line inside the circle that connects two points of a circle’s circumference. A chord is always > arc
• Tangent line: A line outside the circle that intersects ONLY ONE point of a circle’s circumference. If you draw a line from the circle’s center to the intersection point of the tangent line, the angle is always 90°.
• Central angle: Angle formed from the center of the circle
• Inscribed angle: Angle formed from a point on the circumference

Question 24

Circumference

Answer 24

Circumference = π x diameter

Question 25

Central Angle Theorem

Answer 25

The central angle of any arc in a circle is 2 x inscribed angle.

Question 26

Length of an Arc ## Footnote What is the length of the arc with a central angle of 75° and side of 6?

Answer 26

Central Angle ÷ 360° = Arc ÷ Circumference We divide the central angle ÷ 360° b/c an entire circle's circumference/angle corresponds to 360° ## Footnote 1. Set up equation: 75° ÷ 360° = Arc ÷ (6 x 2) π 2. Arc = 2.5π

Question 27

Area of a Circle

Answer 27

Area = π x radius² (diameter)

Question 28

Area of a Sector ## Footnote What is the area of a sector with a central angle of 60° and a side of 10?

Answer 28

Angle of Sector ÷ 360° = Sector Area ÷ Area of circle ## Footnote 1. Set up equation: 60° ÷ 360° = Sector Area ÷ (10)² π 2. Sector Area = 16.6π

Question 29

Inscribed/Circumscribed Polygons

Answer 29

* Inscribed Polygon: A polygon that is inside a circle where all of the polygon's vertices sit on the circumference of the circle. * Circumscribed Polygon: A polygon surrounding a circle so that each of its sides touches the circumference of the circle.

Question 30

Max Area of an Inscribed Polygon

Answer 30

The Max Area of an Inscribed Polygon is its regular shape. To maximize the area of an inscribed square in another square, you have to make the inscribed square look more and more like the original square. (Turn it upright) This means that the SMALLEST possible area for a square inscribed in another square is when the inscribed square's vertices are at the midpoints of the sides of the larger square.

Question 31

Concentric Circles

Answer 31

Two (or more) circles who share the same center. ◎

Question 32

Volume of a Cube/Rectangular Solid

Answer 32

* Cube: Side³ * Rectangular Solid: Height x Length x Width

Question 33

Surface Area of a Cube/Rectangular Solid

Answer 33

* Cube: Side² x 6 * Rectangular Solid: 2 (Height x Length) + 2 (Height x Width) + 2 (Length x Width)

Question 34

Volume of a Right Cylinder

Answer 34

Radius² x π x Height (Area of circle x Height) ## Footnote imagine you have the area of a flat circle ( r²π) stacked on top of itself to a certain height.

Question 35

Surface Area of a Right Cylinder

Answer 35

* (Area of 2 bases) + (Height x Circumference) * (2 x Radius² x π) + (2 x Radius x π x Height)

Question 36

Surface Area/Volume Relationship

Answer 36

* Cube: If the side is <6, Surface Area > Volume 　　　　If the side is > 6, Surface Area < Volume * Cylinder: If the radius is <4, Surface Area > Volume 　　　　　If the radius is > 4, Surface Area < Volume

Question 37

Longest Diagonal of a Cube/Rectangular Solid ## Footnote * What is the longest diagonal of a cube w/ side length of 5? * What is the longest diagonal of a rectangular solid w/ length of 8, width of 9 height of 12?

Answer 37

The longest diagonal of a cube is the point from the front right bottom corner to the back left corner of the cube. * Length of Cube diagonal = Side x √3 * Length of Rectangular Solid diagonal = √(Height² + Length² + Width²) ## Footnote * 5√3 * √(8² + 9² + 12²) = √289