Plane vs. Solid Geometry Plane GeometrySolid Geometry

Plane Geometry vs Solid Geometry Plane GeometrySolid Geometry Two-Dimensional - 2DThree-Dimensional - 3D

Plane Geometry Notes

LINE A STRAIGHT PATH OF POINTS WITH NO ENDPOINTS. A B AB

SEGMENT PART OF A LINE WITH 2 ENDPOINTS A B AB

Lines can be said to have three relationships with each other. Lines Intersecting Skew Parallel Perpendicular

Intersecting Lines – Lines that meet at a point. –Perpendicular Lines lines that intersect to form 90 degree angles. m n M N

Parallel Lines – Lines in the same plane that do not intersect. m n M N

Parallel Lines – Lines in the same plane that do not intersect. Skew Lines - Lines in different planes that do not intersect.

RAY PART OF A LINE WITH ONLY ONE(1) ENDPOINT A B AB

ANGLE TWO RAYS WITH A COMMON ENDPOINT VERTEX K C J <KJC or <CJK

TYPES OF ANGLES ACUTE RIGHT OBTUSE Less than 90° Exactly 90° Greater than 90°

Classify these ANGLES ACUTE RIGHT OBTUSE

End of Section 1

Plane vs. Solid Geometry Plane GeometrySolid Geometry

Today we will look at some angle relationships! Vertical Angles Adjacent Angles Complementary Angles Supplementary Angles It takes two to tango!

Special Angle Pairs: 1) Vertical Angles – the angles formed on opposite sides of intersecting lines <2 and <4 are vertical angles so we know m<2 = m<4 Vertical angles are always congruent

2) Adjacent Angles – are angles that share a common vertex and a common side

Complementary Angles – two angles whose sum is 90 degrees Supplementary Angles – two angles whose sum is 180 degrees

Complementary Angles 60 X = X 30 = X

What is the complement of a 72 degree angle? What is the complement of a 35 degree angle? 18 degrees 55 degrees

Supplementary Angles 38 X X = X = 142 degrees 360° 180°

See IXL.com Q1 and Q2

How many pairs of supplementary angles can you name ?

Angle Facts Starter: Name these angles: Acute Reflex Right Angle Obtuse

90 Vertical Horizontal 360 o Angles at a Point

a b c d Angles at a point add to 360 o Angle a + b + c + d = Angles at a Point

75 o 85 o 80 o a Example: Find angle a o Angle a = ( ) = = 120 o Angles at a Point

a b c d Opposite Angles are equal Angle a + b = because they form a straight line Angle c + d = because they form a straight line Angle c + b = because they form a straight line Angle d + a = because they form a straight line So a = c and b = d Angles at a Point

Horizontal line Oblique line a b Angles a + b = 180 o 70 o b Angle b = 180 – 70 = 110 o x 35 o Angle x = 180 – 35 = 145 o 90 Angles on a straight line add to 180 o 180 o Angles on a Line

Angle Facts Now do these: f g g g 2h h 60 o 45 o 120 o 110 o 35 o a b 22 o 116 o c d e 135 o 80 o 148 o i a = 180 – 35 = 145 o b = 180 – (22+90) = 68 o Opposite angles are equal So c = 116 o d = 180 – 116 = 64 o e = 360 – (135+80) = 145 o f = 360 – ( ) f = = 85 o 3g = 360 – (90+60) = 210 g = 70 i = = 32 o 3h = 180 – i = 148 h = 49.3.

Parallel lines remain the same distance apart. Transversal Draw a pair of parallel lines with a transversal and measure the 8 angles. Vertically opposite angles are equal. Corresponding angles are equal. Angles between Parallel Lines

Parallel lines remain the same distance apart. Transversal Draw a pair of parallel lines with a transversal and measure the 8 angles. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

Parallel lines remain the same distance apart. Transversal Draw a pair of parallel lines with a transversal and measure the 8 angles. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

Parallel lines remain the same distance apart. Transversal a b c e g h f Name an angle corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

Parallel lines remain the same distance apart. Transversal a b c h g d f Name an angle corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

Parallel lines remain the same distance apart. Transversal a b e h g d f Name an angle corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Angles between Parallel Lines

Parallel lines remain the same distance apart. Transversal a b e h g d f Name an angle alternate interior to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Angles between Parallel Lines

Parallel lines remain the same distance apart. Transversal a b e h g d c Name an angle alternate to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Angles between Parallel Lines

Parallel lines remain the same distance apart. Transversal a b e h g d c Name an angle interior to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Angles between Parallel Lines

Parallel lines remain the same distance apart. Transversal a b e h g d c Name an angle interior to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Angles between Parallel Lines

a f e h g d c Name an angle corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Angles between Parallel Lines

a f e b g d c Name an angle alternate to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Angles between Parallel Lines

h f e b g d c Name an angle interior to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

h f e b g d c Name in order, the angles that are alternate, interior and corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

a f e h g d c Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Name in order, the angles that are alternate, interior and corresponding to the marked angle. Angles between Parallel Lines

x Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Finding unknown angles 100 o Find the unknown angles stating reasons, from the list below. y z 60 o  x =  y =  z = 80 o Int.  s 60 o vert.opp.  s 120 o Int.  s Angles between Parallel Lines

x Vertically opposite angles are equal.vert.opp.  s Corresponding angles are equal.corr.  s Alternate angles are equal.alt.  s Interior angles sum to 180 o.(Supplementary) Int.  s Finding unknown angles 105 o Find the unknown angles stating reasons, from the list below. y z  x =  y =  z = 105 o corr.  s 55 o alt.  s 125 o Int.  s 55 o Angles between Parallel Lines

x Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Finding unknown angles 95 o Find the unknown angles stating reasons, from the list below. y  x =  y = 85 o Int.  s 120 o Int.  s 60 o Unknown angles in quadrilaterals and other figures can be found using these properties. Angles between Parallel Lines

x Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Finding unknown angles Find the unknown angles stating reasons, from the list below. y 55 o z  x =  y =  z = What does this tell you about parallelograms? 125 o Int.  s 125 o 55 o Int.  s 55 o 125 o Int.  s 125 o Unknown angles in quadrilaterals and other figures can be found using these properties. Angles between Parallel Lines

70 o a Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Find the unknown angles stating reasons, from the list below. There may be more than one reason. 58 o vert.opp.  s 32 o  s in tri 58 o 32 o alt.  s b d c Angle sum of a triangle (180 o ) Angle on a line sum to (180 o ) 58 o  s on line e 58 o corr.  s 52 o  s at a point fg h Base angles isosceles triangle equal. 64 o isos tri  a =  b =  c =  d =  e =  f =  g =  h = 64 o isos tri Angles at a point sum to 360 o Mixing it! Angles between Parallel Lines

Angle Facts Now do these: 65 o p 99 o 77 o 38 o 54 o 48 o 35 o z 130 o y x w t u s r q v Corresponding angles p = 65 o Alternate angles q = 38 o Corresponding angles r = 77 o Opposite angles (with r) or Alternate angles with 77 o s = 77 o t = 99 o v = 54 o u = 81 o w = 126 o x = 130 o y = 130 o z = 35 o + 48 o = 83 o

POLYGON A CLOSED PLANE FIGURE MADE UP OF THREE OR MORE LINE SEGMENTS

NAMES OF DIFFERENT POLYGONS TRIANGLE - 3 SIDES QUADRILATERAL - 4 SIDES PENTAGON - 5 SIDES HEXAGON - 6 SIDES

HEPTAGON - 7 SIDES OCTAGON - 8 SIDES NONAGON - 9 SIDES DECAGON - 10 SIDES

HENDECAGON - 11 SIDES DODECAGON - 12 SIDES TRISKAIDECAGON - 13 ICOSAGON - 20 SIDES

Triangle Polygon with three sides

Congruent – having equal measure This could be angles, lines, or polygons.

Classifying Triangles by the SIDES SCALENE No Congruent Sides ISOSCELES Two Congruent Sides EQUILATERAL Three Congruent Sides

Classifying Triangles by the ANGLES ACUTE All Angles Less Than 90° RIGHT Has one Right Angle OBTUSE Has one Obtuse Angle

The SUM of the measures of the angles of a triangles is always180° = 180° = 180°

Homework was p Acute 20. Right 21. Right

REGULAR POLYGON POLYGONS THAT HAVE EQUAL SIDES AND ANGLES

Classwork/Homework is p (1-21) Begin now

Special Quadrilaterals A TRAPEZOID has only one set of parallel lines A PARALLELOGRAM has two sets of parallel lines.

Special PARALLELOGRAMS A RECTANGLE has four right angles A RHOMBUS has 4 congruent sides. A SQUARE has four 90° angles and 4 congruent sides

Parallelogram Facts to Know: Opposite sides are the same length Opposite angles are the same measure.

The SUM of the Measures of the Angles of a Quadrilateral is 360° ° = 360°

To find the sum of the angles of any polygon you can use the formula 180(n-2), where n is the number of sides of the polygon.

CONGRUENT POLYGONS POLYGONS THAT HAVE THE SAME SHAPE AND THE SAME SIZE

SIMILAR POLYGONS POLYGONS WITH THE SAME SHAPE BUT NOT THE SAME SIZE

SIMILAR POLYGONS Corresponding Angles are Equal

SIMILAR POLYGONS Sides are in proportion 20 X 24 16

LINE OF SYMMETRY A LINE THAT DIVIDES A FIGURE INTO TWO CONGRUENT PARTS THAT WILL FLIP TO MATCH UP.

COORDINATE PLANE - A grid formed by an X-axis running left to right and label like a number line including both positive and negative numbers; and a Y-axis running up and down and labeled like a number line including positive and negative numbers. The ORIGIN is the point both number lines equal zero. (0,0)

Coordinates are written in the form (x,y) where x is the value of the x-axis and y is the value of the y- axis.

The grid is separated into 4 areas known as QUADRANTS: Quadrant I has a positive X and a positive Y. Quadrant II has a negative Y and a positive Y Quadrant III has a negative X and a negative Y. Quadrant IV has a positive X and a negative Y

Quadrant I - (+,+) Quadrant II - (-,+) Quadrant III - (-,-) Quadrant IV - (+, -) In other words...

Finding Areas... Remember: Area of a rectangle is found by A = L x W L = 22 cm W = 10 cm So, A or Area = L x W A = 22 x 10 A = 220 cm 2

To find area of a triangle... H = 14 B = 12 cm Think of a triangle as 1/2 of a rectangle. So its area is 1/2 of the B x H. A = 1/2 x B x H A = 1/2 x 12 x 14 A = 6 x 14 A = 84 cm 2

Sample: Find the sum of the angles of a octagon. An octagon has 8 sides, so if we use the formula, 180(n-2); 180 (8-2) 180 x ° total in an Octagon