Math.docx

4th Quarter – MATH Geometry – came from the greek word “geo” and “metria” which means earth & measure  Point • A  Line Through any two points there exists exactly one line & A line contains at least two points.  Ray - A line that can be extended in only one direction - Starts with an end point - Opposite rays – they both have the same end points  Line segment - Has two end points which measure is fixed  Plane - Can be extended in all sides  Parallel Lines -lines that never meet - if the distance between them is equal (equidistant) - symmetrical  Perpendicular Lines -intersecting lines forming 90° right angle -If two lines are perpendicular, then they meet to form right angles -two lines that meet to form congruent adjacent angles  Angle – formed when two lines intersects  Acute Angle – less than 90°  Right Angle – exactly 90°  Obtuse Angle – more than 90° but less than 180°  Straight Angle – anne mira delos santos and exactly 180°  Reflex Angle – more than 180° and less than 360° -

Naming an Angle  Letters including the vertex ∟  ABC  Vertex ∟B  Numbers written in the angle Angle Pairs  Complementary Angles – the sum of two angles is equal to 90°  Supplementary Angles – the sum of two angles is equal to 180°  Vertical Angles - If two angles are vertical angles, then they are congruent -each of the pairs of opposite angles made by two intersecting lines  ∟1&3 ∟2&4 ∟8&6 ∟7&5  Same-Side Interior Angles – pair of angles on one side of a transversal line, and on the inside of the two lines being intersected -same side interior angles are supplementary  ∟4&8 ∟3&5

 Same-Side Exterior Angles - both angles are on the same side of the transversal line, and exterior tells us that both angles are exterior, or outside, of the parallel lines. - same-side exterior angles are supplementary  ∟1&7 ∟2&6  Alternate Interior Angles - pairs of angles on opposite sides of the transversal but inside the two lines - If two parallel lines are cut by a transversal, then alternate interior angles are congruent • ∟4&5 ∟3&8  Alternate Exterior Angles - pairs of angles on opposite sides of the transversal but outside the two lines - If two parallel lines are cut by a transversal, then alternate exterior angles are congruent • ∟1&6 ∟2&7  Corresponding Angles - If two parallel lines are cut by a transversal, then the corresponding angles are congruent - The angles in matching corners • ∟1&8 ∟2&5  Linear Pair - If two angles form a linear pair, then the measures of the angles add up to 180° - a pair of adjacent, supplementary angles, adjacent means next to each other • ∟1&2 ∟8&5  Polygons – close figure  Convex polygon- no diagonal goes outside the figure as it travels from one corner to the other. - Another property of convex polygons is that no angle inside the polygon will have a measure greater than 180 degrees. -A convex polygon has all its vertices, or corners, pointing out from the center  Concave polygon - at least one diagonal passes outside the figure. - at least one angle inside the polygon will have a measure greater than 180 degrees

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a concave polygon looks like it has been caved in Quadrilaterals – shapes that are four-sided

 Quadrilateral - If there are four points on a plane and no three of these points are collinear and the segments only intersect at their endpoints then the union of these segments - The points are called vertex/vertices, the segment are called sides , the angles are called angles - Convexity is an important property of quadrilaterals - A quadrilateral is convex if no two of its vertices lie on opposite sides of a line containing a side of the quadrilateral - The sum of the measures of the interior angles are 360° - Two sides of a quadrilateral are opposite if they do not intersect , two sides of its angles are opposite if they have no common side - Two sides are consecutive if they have a common endpoint. Two angles are consecutive if they have a side in common - A diagonal of a quadrilateral is a segment joining two non-consecutive vertices

FAMILY OF QUADRILATERALS :

QUADRILATERALS

Parallelogram

Rhombus

Trapezoid

Rectangle

Trapezium

Isosceles Trapezoid

Square

 Parallelogram - A quadrilateral in which both pairs of opposite sides are parallel and congruent - Diagonals bisect each other - two opposite angles are congruent - the two opposite sides are congruent - the two consecutive angles is equal to 180° -each diagonal separates a parallelogram into two congruent triangles -two consecutive interior angles are supplementary

Parallelogram- Properties  Rectangle  Square - A parallelogram -is a rectangle all of whose angles are all whose sides are right angles congruent -diagonals of a -diagonals of a rectangle are square are congruent congruent -diagonals of a square are perpendicular to each other

 Rhombus - A parallelogram with four congruent sides - Diagonals of a rhombus bisects the angles whose vertices are its endpoints - Diagonals of a rhombus are perpendicular to each other

Theorems  If both pairs of opposite sides of a quadrilateral are congruent ,then the quadrilateral is a parallelogram  If a pair of opposite angles of a quadrilateral are both parallel and congruent ,then the quadrilateral is a parallelogram  If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

Finding x and y Finding measures

Parallelogram & Retangle

x braces are also used to provide support in a rectangular fencing, If AB=6 ft , AD=2ft and m∟DAE =65. Find each measure.

X+2y 2x-3y

15 25

1.Equate

4.m∟EDC=115 (65+65=130 , 360130=230÷2 =115)

2x+4y = 50 7𝑦=35 7

(2x+4y=50)-(2x-3y=15)=

X=25-10 x=15

4.m∟EDC

3. If AB=6ft then, DB=6ft

2(x+2y)=(25)(2)

X+10=25

2.m∟CEB

2. ∟A = 65 (∟DAE=65) ;; ∟CEB = 65

2. Multiply to Eliminate

X+2(5)=25

3.DB

1.AD =2ft ;; BC=2ft

X+2y=25 ;; 2x-3y=15

3. Substitute

1.BC

= y=5

Trapezoid  A quadrilateral which has one and only one pair of parallel sides they are denoted by arrow heads  Parallel sides are calles bases , the non-parallel sides are called legs, the segment joining the midpoints of the legs is called median. Angles whose vertex are bases is calles base angles  Legs are congruent  Consecutive angles between the bases of a trapezoid are supplementary  When the non-parallel sides of a trapezoid have the same length its called isosceles trapezoid.  The base angles of an isosceles trapezoid are congruent and the Diagonals of an isosceles trapezoid are congruent  If the base angles of an isosceles trapezoid are congruent and the Diagonals of an isosceles trapezoid are congruent therefore it’s an isosceles trapezoid  Isosceles Trapezoid – non-parallel sides of the trapezoid have the same length  Median of a Trapezoid – segment which connects the midpoints of the non-parallel sides, it’s parallel to the base 1 𝑀𝑒𝑑𝑖𝑎𝑛 = (𝑏1 + 𝑏2 ) 2 Trapezium  You cannot see any type of a parallel side

Kites       

A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not. Has no parallel sides an example of trapezium but not a parallelogram or a trapezoid If a quadrilateral is a kite then its diagonal are perpendicular Consecutive congruent sides Diagonals bisect the non-congruent angles If a quadrilateral is a kite then exactly one pair of opposite angles are congruent

Proving

Similarity- the same shapes but differ in sizes  Similar Triangles – congruent angles - Proportional sides  Ratio – comparison between two quantities - An expression of comparisom between two quantities by division  Proportion – equally between two ratio - Reflexive Property: For any real number a, a=a - Symmetric Property: If a=b = , then b=a Transitive Property: If a=b and b=c , then a=c - The numbers in a proportion are called terms of the proportion - The first and last terms are extremes

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The second and third terms are means Cross Product Property – the product of them extremes is equal to the product of the means Reciprocal Property- the terms of proportion are in proportion by inversion 𝑎

If a,b,c are positive numbers and 𝑏 =

𝑏 𝑐

then b=√𝑎𝑐

Similar Polygons  Similar Polygons – two polygons are similar if corresponding angles are congruent and lengths of corresponding sides are proportional  To Identify if two polygons are similar:  Identify thr corresponding vertices  Find if corresponding angles are congruent  Determine if the lengths of corresponding sides are proportional.  Finding the missing term :  Identify the proportional sides  Substitute with the terms  Cross-multiply  Division Property of Equality Proving Similar Triangles  AA Similarity – two pairs of congruent angles then two triangles are similar  SSS Similarity Theorem – three sides are proportional therefore the two triangles are similar  SAS Similarity Theorem – If two pairs of corresponding side are proportional, and the included angles are congruent , then the correspondence is a similarity

 BPT (Basic Proportionality Theorem) – If a line parallel to one side of a triangle intersects the other two sides of a triangle in distinct points, then it cuts off segments which are proportional to these sides - If a line intersects two sides of a triangle and cuts off segments proportional to these sides then it is parallel to the third side  Angle Bisector Theorem – the bisector of an triangle seperates the opposite sides into segments whose lengths are proportional to the lengths of the other two sides. Pythagorean Theorem – In a right triangle , the square of the length of the hypotenuse is sum of the squares of the lengths of the legs C2 = a2+b2

Problem solving involving similar polygons