Angles (Sum of Angles)

First

The figure below consists of 3 straight lines. What is the value of  ∠a + ∠b + ∠c?

Method 1

Using Vertically Opposite Angles, the angle in between ∠a & ∠c is equal to ∠b.

Using Adjacent Angles on a Straight Line, ∠a + ∠b + ∠c = 180° because ∠a, ∠b & ∠c lie on a straight line

Method 2

Using Vertically Opposite Angles, the other 3 angles are equal to ∠a + ∠b + ∠c.

Using Angles at a Point:

2 (∠a + ∠b + ∠c) = 360° or 2∠a + 2∠b + 2∠c = 360°

Therefore, ∠a + ∠b + ∠c = 360° / 2 = 180°

Second

The figure below is not drawn to scale. It is made up of 4 identical squares. find the sum of the 3 marked angles.

Label the topmost angle as a, the one directly beneath is as b and the one on its right as c.

b = 45 (angle bisector of a square)
a + c = 90 (double alternate angles)
a + b + c = 45 + 90 = 135


Third
The figure below is not drawn to scale. It shows two triangles inside a square ABCD. AE = CD. Find ∠BED.

Method 1 (Trans-Diagram)

Extend line AE upwards. Cut each of the two new exterior angles created into halves. Each of the exterior angle halves is equal to the interior opposite base angles of its own isosceles triangle.

(360 - 90) / 2 = 135 (angles at point A)

Method 2 (Algebraic)

AED = (180 - DAE) / 2

AEB = (180 - BAE) /2 

DAB =DAE + AEB = 90

AED + AEB 

= (180 - DAE) / 2 + (180 - BAE) /2 

= (360 - DAE - BAE) / 2

= (360 - DAB) / 2

= (360 - 90) /2

= 135

Circles (Terminology)

Circumference
= Line Around a Circle
= Perimeter

Circumference is slightly more than 3 times the diameter in a ratio given by Pi which is approximately 3.14 or 3 1/7

Centre of a Circle is a point that is of equal distance from every point on the circumference.

The Diameter is any line that starts from one point on a Circumference, passes through the Centre and ends on another point on the opposite side of a circumference.

The Radius is any line that starts from the Centre and ends on any point on the Circumference or vice versa. It is half the diameter.

A Chord is any line that starts from one point of a Circumference and ends at another point on the same Circumference. Diameter is a special type of Chord.

A Tangent is any line that starts and ends from outside the Circle but touches it only at one point.

A Secant is any line that starts and ends from outside the Circle and cuts the Circle at 2 points. It is like a Chord but with a line that extends betond the Circumference. A Tangent line is a special type of Secant.

A Semicircle is Half a Full Circle with a Central Angle of 180°. It's Area and Arc Length is also Half that of a Full Circle. It

A Quadrant is a Quarter of a Full Circle with a Central Angle of 90°. It's Area and Arc Length are also a Quarter of a Full Circle.

A Sector is a slice of a Full Circle. The Semicircle and the Quadrant are special types of Sectors.

An Arc is a portion of a Circumference.

Ordered Range Concept

The number of things between two ordinal numbers is equal to the difference and 1.

Ordered Range = Difference + 1

Taking a Difference eliminates the first object on an Ordered Range. Adding 1 restores that object back.

Derived from:

Elimination Concept
Difference Concept

Difference Concept

The Difference is gap between two things. The nearer they are, the smaller the difference. The further apart they are, the wider the difference.

Difference = Big Part - Small Part

The Small Part is separated from the Big Part by a Difference. To find something smaller from a bigger thing, we must Subtract.

Small Part = Big Part - Difference

The Big Part is separated from the Small Part by a Difference. To find something bigger from a smaller thing, we must Add.

Big Part = Small Part + Difference

Derived From:
Elimination Concept
Small to Big Concept
Big to Small Concept

Application:

1) Algebra

Total Concept

The Total is a Sum of its Parts. It is bigger than every Part within it. To find something bigger from smaller quantities, we have to Add.

First Part + Second Part = Total

Each Part is smaller than the Total. To find something smaller from a bigger quantity, we have to Subtract.

First Part = Total - Second Part
Second Part = Total - First Part

Application:

1) Geometry
- Angle Sum of Triangle
- Adjacent Angles on a Straight Line
- Angles about a Point
- Pythagoras' Theorem
2) Algebra
3) Arithmetic
4) Trigonometry

Etc

This concept is derived from the Small to Big Concept and the Big to Small Concept

Trigonometry (Pythagorean Identity)

(sin A)^2 + (cos A)^2 = 1
Square of Height + Square of Base = Square of Slope

Using a Unit Circle with Radius = 1, with a Centre about the Origin (0,0) on the Cartesian Plane, a perpendicular line drawn from any point on the circle's circumference to the X-Axis produces a right angle.

That perpendicular line (height of the triangle) is Opposite to the Angle between the X-Axis and the hypotenuse. It represents the values about the Y-Axis. Since Hypotenuse = 1, sine of angle equals the y values.

The base of the triangle is along the X-Axis and thus represents the values about the X-Axis. It is Adjacent to the Angle. Since Hypotenuse = 1, cosine of angle equals the x values

The slope of the triangle which has a value of 1 which is also the radius of the Unit Circle, is the Hypotenuse. It is opposite to the right angle of the triangle.

Using Pythagoras' Theorem,

Y^2 + X^2  = 1^2
(sin A)^2 + (cos A)^2 = 1

Circles Concepts (Difference in Area)

The diagram below shows a circle with center O. AB and XY are diameters of the circle. Find the difference between the shaded and unshaded areas