Geometry Basics: Angles and Triangles

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Questions and Answers

What does one complete revolution equal?

180 degrees

90 degrees

360 degrees (correct)

270 degrees

What are the degrees of an acute angle?

less than 90 degrees

What are the degrees of a right angle?

90 degrees

What are the degrees of an obtuse angle?

greater than 90 degrees Signup and view all the answers

What are the degrees of a straight angle?

180 degrees Signup and view all the answers

If two angles have a sum of 90 degrees, what are they called?

complementary angles Signup and view all the answers

If two angles have a sum of 180 degrees, what are they called?

supplementary angles Signup and view all the answers

What is the equation for the complementary angle?

90 - y Signup and view all the answers

What is the equation for the supplementary angle?

180 - y Signup and view all the answers

What does a little line (tick) mean if it's going through a line of a triangle?

indicates sides are different in length Signup and view all the answers

Where are the legs a, b, hypotenuse c, and theta located in a right triangle?

a is bottom, b is left, c is longest side, theta is inside Signup and view all the answers

What is the equation for the Pythagorean theorem?

a^2 + b^2 = c^2 Signup and view all the answers

What are the degrees for a special triangle that has equal sides?

30 degrees, 60 degrees, 90 degrees Signup and view all the answers

What is the ratio of a 30-60-90 triangle?

t : sqrt(3) * t : 2 * t Signup and view all the answers

What are the two types of special triangles in degrees?

30-60-90 and 45-45-90 Signup and view all the answers

What is the ratio for a 45-45-90 degree triangle?

t : t : sqrt(2) * t Signup and view all the answers

What is the vertex of a square?

the common point Signup and view all the answers

Where are the first, second, third, and fourth quadrants located on a graph?

Bottom left- Negative x, Negative y Signup and view all the answers

What is the distance formula?

sqrt((x2 - x1)^2 + (y2 - y1)^2) Signup and view all the answers

What letters signify the center point and what letter signifies the radius when finding the equation of a circle?

(h,k) and r Signup and view all the answers

What are the trigonometric functions?

All of the above Signup and view all the answers

What formula do we use to find the remaining trigonometric functions?

r^2 = x^2 + y^2 Signup and view all the answers

The reciprocal identities include 1/cosθ = secθ.

True Signup and view all the answers

What are the ratio identities?

cotθ = x/y = cosθ/sinθ, tanθ = sinθ/cosθ Signup and view all the answers

What is the rule to memorize the signs of trigonometric functions in a given quadrant?

CAST rule Signup and view all the answers

Study Notes

Basic Angle Measurements

One complete revolution is 360 degrees.
Acute angles measure less than 90 degrees.
Right angles measure exactly 90 degrees.
Obtuse angles measure greater than 90 degrees but less than 180 degrees.
Straight angles measure 180 degrees.

Angle Relationships

Complementary angles sum up to 90 degrees.
Supplementary angles sum up to 180 degrees.
Complementary and supplementary formulas: 90 - y and 180 - y respectively.

Triangle Properties

Tick marks on triangle sides indicate length differences (one tick means equal length, different ticks indicate varied lengths).
In a right triangle:
- Hypotenuse (c) is the longest side.
- Leg a is the bottom side, leg b is the left side.
- Theta (θ) is located at the angle inside the triangle.

Pythagorean Theorem

The formula is a² + b² = c², where c is the hypotenuse.

Special Triangles

Equilateral triangles have three equal sides and angles: 60°, 60°, 60°.
A 30-60-90 triangle has angles of 30°, 60°, and 90° with side ratios t : √3t : 2t.
A 45-45-90 triangle has angles of 45°, 45°, and 90° with side ratios t : t : √2t.

Graphing Coordinates

First Quadrant (top right): x and y are positive.
Second Quadrant (top left): x is negative, y is positive.
Third Quadrant (bottom left): x and y are negative.
Fourth Quadrant (bottom right): x is positive, y is negative.

Distance and Circle Formulas

Distance formula: √[(x₂ - x₁)² + (y₂ - y₁)²].
Circle equation center is at (h,k) with radius r: (x - h)² + (y - k)² = r².

Trigonometric Functions

Sine: sin(θ) = y/r
Cosine: cos(θ) = x/r
Tangent: tan(θ) = y/x
Secant: sec(θ) = r/x
Cosecant: csc(θ) = r/y
Cotangent: cot(θ) = x/y.

Finding Trigonometric Functions

To find remaining trig functions, use r² = x² + y² (derived from the Pythagorean theorem).

Identities in Trigonometry

Reciprocal identities:
- cscθ = 1/sinθ
- secθ = 1/cosθ
- cotθ = 1/tanθ
Ratio identities:
- cotθ = x/y = cosθ/sinθ
- tanθ = sinθ/cosθ.

CAST Rule

Used to memorize signs of trigonometric functions in different quadrants:
- C (cos) in the fourth quadrant is positive.
- A (All) in the first quadrant means all functions are positive.
- S (sin) in the second quadrant is positive.
- T (tan) in the third quadrant is positive.

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Description

Test your knowledge on basic angle measurements and properties of triangles. This quiz covers the types of angles, relationships between them, and the Pythagorean theorem. Perfect for students studying geometry or anyone looking to refresh their math skills.