Indirect Proofs

Indirect Proofs:

-It is when the proof is already assumed to be false, and a contradiction must be proven.

To write an indirect proof, use the following steps:

1. Identify the conjecture to be proven.

2. Assume the opposite (the negation) of the conclusion is true.

3. Use direct reasoning to show that the assumption leads to a contradiction.

4. Conclude that since the assumption is false, the original conjecture must be true.

Perpendiculater and Angle Bisectors

Perpendicular Bisectors:

A term needed to know is equidistant, when talking about perpendicular bisectors.

-Equidistant: This is when a point is the same distance from two or more objects.

Distance and Perpendicular Bisectors:

1. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of a segment.

2. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Angle Bisectors:

Distance and Angle Bisectors:
1. If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
2. If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.

Sector Area and Segment Area

Sector of a Circle:

-A sector of a circle is a area bound by two radii of a circle and their intercepted arc.

Area:

-To find the area of a sector, use the following equation:

Segment of a Circle:

-A segment of a circle is an area bound by an arc with its chord.

To find the area of a segment, first:

1. Find the area of the sector.

2. Find the area of the triangle: A=1/2bh

3. Find the difference between the area of a sector and the area of a triangle.

4. This is the area of a segment.



  

Kites and Trapezoids

Kites:

-A kite is a quadrilateral that has two pairs of congruent sides.

Properties of Kites:

1. A kite's diagonals are perpendicular

2. Only one pair of opposite angles are congruent.

Trapezoids:

-A trapezoid is a quadrilateral with only one pair of parallel sides.

Terms of Trapezoids:

1. Base: Each of the parallel sides.

2. Leg: The nonparallel sides.

3. Base Angles: Two consecutive angles whose common side is a base.

Isosceles Trapezoids:

1. Each pair of base angles are congruent.

2. Is isosceles if has one pair of base angles.

3. Diagonals are congruent.

Quadrilaterals are Everywhere!

A quadrilateral is a four-sided shape that when split in half a certain way, it contains two triangles.

Some examples to prove that we live in a place where quadrilaterals are everywhere are:

Windows

Doors

Pictures and Picture Frames

Crates

Circles Amazing?

Yes, circles are in fact very amazing shapes. 

First, there are four different lines and segments that intersect circles. They are:

1. Chord: This is a segment whose two endpoints lie on a circle.

2. Secant: This is a line that intersects a circle at any two points.

3. Tangent: This is a line that intersects a circle at only one point.

4. Point of Tangency: This is the point where a tangent intersects a circle.

Also, when learning about triangles, the terms radius and diameter should be known.

Radius: This is a segment that spreads from the exact point in the middle of a circle to the circle itself.

Diameter: This is a segment that splits the circle exactly in half, and it is also twice the size of the radius.