Geometry 47 Line notation and relationships 46 Measure, draw, and name angles 45 Angle Pairs 44...
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Transcript of Geometry 47 Line notation and relationships 46 Measure, draw, and name angles 45 Angle Pairs 44...
Geometry
47 Line notation and relationships46 Measure , draw, and name angles45 Angle Pairs44 Identify polygons and quadrilaterals43 Solids42 Translation, rotation, reflection, and dilations41 Missing angles in triangles40 Classify s and s39 Similarity and Congruence38 Similar triangles and proportions37 WP: Similar triangles and proportionsIce Cream Answer
Puzzle - PresidentThe 22nd and 24th president of the United States had the same mother and the same father, but they were not brothers. How is this possible?
Who was this???
47a Line notation and relationships
Use the figure to name a line containing point K.
Answer: The line can be named as line a.
There are three points on the line. Any two of the points can be used to name the line.
A line can be named by a lower case letter or by two points on that line.
47b Line notation and relationships
Ray
Line
Means Perpendicular (90°)
Means “Line” (Has No Endpoints)
Means “Ray” (Has one endpoint)
— Means “Line Segment” (Has two endpoints – Has a length)
Means “Perpendicular” (Meets in a 90° angle or Right )
|| Means “Parallel” (Follow same direction – Never intersect)
Skew lines are in different planes and never intersect
***If you have a protractor, bring it to class!***
If a daddy bull weighs 1,200 pounds and eats 12 bales of hay each day, and a baby bull, who weighs 300 pounds, eats 3 bales of hay each day, then how much hay should a mommy bull eat if she weighs 800 pounds?
Puzzle - Know bull
46a Measure, draw, and name angles
An angle is usually named by 3 noncollinear points
Angles can be named with three points on the angle with the vertex in the middle or by the vertex point if it is the only angle with that vertex.
L
M N
LMN or NML or M are all correct
MLN or L etc are NOT correct
When referring to an angle you may also mark it with an arc.
1
An angle may also be named by a number such as above. 1
46b Measure, draw, and name angles
Acute – Less than 90°
Right – Exactly 90° – Marked by a Small
Obtuse – Greater than 90° but less than 180°
Straight – Exactly 180°
Complementary s – Two angles that add up to 90° (Right)
Supplementary s – Two angles that add up to 180° (Straight)
Adjacent s – Two angles with a common sideand a common vertex
A complete circle = 360°
46c Measure, draw, and name angles
Measure each angle named and classify it as right, acute, or obtuse.
a. CZD
b. CZE
c. DZX
Answer: 150, obtuse
Answer: 90, right
Answer: 30, acute
46d Measure, draw, and name anglesWhat is the measure of the following angle?
Puzzle
What orchestral device is not blown, bowed, plucked, or struck?
45a – Angle pairs
Linear pair – Two angles that form a straight angle (180)Vertical Angles – Angles directly across from each other
when two lines intersect – Vertical angles are congruent ()
Vertical angles
DE
F G
DEF and FEG form a Linear Pair
*Complementary s – Two angles that add up to 90°
*Supplementary s – Two angles that add up to 180°
*Adjacent s – Two angles with a common side And a common vertex
Puzzle - Impossible Triangle
44a Identify polygons and quadrilaterals
3 sides - Triangle
4 sides - Quadrilateral
5 sides - Pentagon6 sides - Hexagon
7 sides - Heptagon or septagon*
8 sides - Octagon
9 sides - Nonagon
10 sides - Decagon
11 sides - 11-gon
12 sides - Dodecagon*
13 sides - 13-gon
Polygon – “Poly” means many, “gon” means angles
Is a circle a polygon? No
*These are the most likely polygons to be named on your final exam….
Know these!!!!!
44b Identify polygons and quadrilaterals
Parallelogram – Both pairs of opposite sides ||
Properties
Opposite sides are Opposite s are
>>
>>> >
44c Identify polygons and quadrilaterals
Rectangle – Quadrilateral with 4 Rt. s
Properties
Opposite sides are - 2 pairs of || sides
44d Identify polygons and quadrilaterals
Rhombus – Quadrilateral with all four sides
Properties
All properties of a parallelogram
Diagonals are
44e Identify polygons and quadrilaterals
Square – Both a rhombus and a rectangle
Properties
Four sides and four s
44f Identify polygons and quadrilaterals
Trapezoid – Exactly ONE pair of || sidesThe || sides are called the basesThe non-|| sides are called legs
Isosceles Trapezoid – One pair of || SidesBoth legs are congruent
||||
Both pairs of base angles are
44g Identify polygons and quadrilaterals
A kite is basically two isosceles triangles
Properties
Diagonals form Rt. Angles
Puzzle
What is it you sit in, sleep in, and brush your teeth with?
43a SolidsPolyhedron = “Poly” means many, “hedron” means faces
Faces = Flat surfaces (polygons)
Edges = Line where two faces meet
Vertices = Point or corner where edges meet
Prism = Has two bases that are ||Named by the shape of the base
Pyramid = Has only one base and comes to a pointNamed by the shape of the base
Is this a polyhedron?
NO
FacesEdgesVertices
612
8
A net is simply a shape unfolded
=
=
43b Solids
43c Solids
43d Solids - ActivityTetrahedron
Faces Vertices Edges
Cube
Faces Vertices Edges
Octahedron
Faces Vertices Edges
Icosahedron
Faces Vertices Edges
Dodecahedron
Faces Vertices Edges
These shapes are called the Platonic Solids.
Why are they special?What to they mean?
Clock Trick
42a Trans, Rot, Refl, & Dilation
Remember that ordered pairs describe a location in terms of Remember that ordered pairs describe a location in terms of x and y such that (x,y).x and y such that (x,y).
These are always in alphabetical order
Where is point A and M located? A (-3,4) M(1,2)
Also recall the quadrant values
II I
III IV
42b Trans, Rot, Refl, & DilationTranslation is simply a movement of a shape
Ex. Points (–8, 4), (–8, 9), (–2, 9), and (–2, 4) form a quadrilateral. Which graph displays the quadrilateral and its dotted translation 5 units to the right and 6 units down?
42c Trans, Rot, Refl, & Dilation
Rotation is simply turning an object
Ex. Draw the rotation image of the figure for a rotation of 270clockwise around turning point D.
42d Trans, Rot, Refl, & DilationReflection simply “flips” or “reflects” an image over a line
42e Trans, Rot, Refl, & Dilation
Dilation is when something changes in sizeThink of your eyes dilating - The pupil changes size
The Scale Factor is by how much it changes size
Ex. The dotted triangle is the image of the solid triangle under a dilation with center (0,0). What is the scale factor?
42f Trans, Rot, Refl, & Dilation
Quiz tomorrow on sections 83-7983 Line notation and relationships82 Measure , draw, and name angles81 Angle Pairs80 Identify polygons and quadrilaterals79 Solids78 Translation, rotation, reflection, and dilations
15 Questions - 15 Points
Puzzle
If I have one and a half dozen kids, how many children do I have.
(By the way, this is not true)
41a Missing angles in a triangle
The sum of the interior angles of a triangle = 180°
A
B
C
A + B +C = 180°
If mA=32 and mB= 105, What is the measure of C?
32 + 105 + C = 180°
The measure of C = 43°
32
105
C = 180 - 32 - 105C = 43
41b Missing angles in a triangle
Exterior
A
B
C
D
E•
•132°
x
Solve for x.
?48°
180-90-48= 42
x=42°
40 Classify s and s
My friends party……
A friend of mine has three kids
Product of their ages is 72
Sum of ages is house number
Oldest child loves strawberry ice cream
How old are his children?What is his house number?
40 Classify s and s
40a Classify s and s
Triangles be classified by the measures of their angles
Acute All angles are less than 90°
Right One angle measures 90 °
Obtuse One angle is greater than 90 °
40b Classify s and sTriangles can be classified by lengths of sides
Equilateral All sides and all angles are
Isosceles Two sides and two angles
Scalene No sides and no angles
Right Lengths a2+b2=c2
(c is the longest side)c
PuzzleCarry’s mother had a rather unusual fixation for naming her four children. The first child was named Quarter, the second child was named Dimeone and the third child was named Nicholas.
What was the fourth Childs name?
39a Similarity and congruence
Similar polygons ( ~ ) have corresponding angles and the corresponding sides are proportional. Same shape, different size.
AB
CD
W
X
YZ
Similarity Statement
ABCD~WXYZ
~
We can conclude:
A WB X
D ZC Y
WX
AB
XY
BC
YZ
CD
ZW
DA
Congruent polygons ( ) are exactly the same shape and size
39b Similarity and congruenceOne triangle is shown on the grid. Two coordinates for a second triangle are also shown on the grid. Using the two points on the graph, which coordinates will create another triangle that is similar to the triangle that is shown?
[A] (–6, –2)
[B] (–6, 5)
[C] (–6, –7)
[D] (–6, –5)
39c Similarity and congruenceWhich two figures are similar but not congruent?
[A] 1 and 5 [B] 3 and 5 [C] 2 and 4 [D] 2 and 3
Triangles BFC and QVU are similar. Write a proportion correctly identifying the corresponding sides?
39d Similarity and congruenceWhich statement is false?
[A] g k [B] a f [C] a f [D] a k
39e Similarity and congruenceCDE ~ HGF . Find the measure of x.
x=60°
Puzzle
Has anyone figured out the strawberry ice-cream question yet?
A friend of mine has three kids
Product of their ages is 72
Sum of ages is house number
Oldest child loves strawberry ice cream
How old are his children?What is his house number?
38a Similar s and proportions
~ Polygons have same shape, and proportional sides.
A
B
CD
W
YZ
~2 3
69
34.5
AD=2 and WZ=3,
If AB=6, what is the length of WX? ___
X
9
If YZ=4.5, what is the length of CD? ____3
38b Similar s and proportionsGiven that ABC ~ DEF, solve for x and y.
If ABC ~ DEF, BC = 3 feet, EF = 12 feet, and FD = 1 foot, find AC.
A
B
CD
E
F
12
1
3
?
38c Similar s and proportions
ABC and XYZ are similar. If AB, BC, and AC are 6 inches, 11 inches, and 15 inches respectively, and XY is 9 inches, find XZ to the nearest tenth.
A
B C Y
X
Z
6
11
15 9
?
Puzzle
Don’t read the words below, just say the color they are printed in.
REDYELLOWGREENBLUERED
BLUEYELLOWGREEN
37a WP: Similar s and proportions
Standing next to each other, a woman casts a 51.2-inch shadow and her 46-inch-tall daughter casts a 36.8-inch shadow. What is the height of the woman to the nearest inch?
51.2
46
36.8
x
37b WP: Similar s and proportionsTwo ladders are leaning against a wall at the same angle as shown. How long is the shorter ladder?
37c WP: Similar s and proportions
Ruby wants to find the height of the tallest building in her city. She stands 399 feet away from the building. There is a tree 32 feet in front of her, which she knows is 15 feet tall. How tall is the building? (Round to the nearest foot.)
37d WP: Similar s and proportions
37e WP: Similar s and proportionsMichele wants to measure the height of the streetlight outside her house. She places a mirror on the ground 46 feet from the streetlight, then walks backward until she is able to see the top of the streetlight in the mirror. Her eyes are 5.4 feet above the ground, and she is 12 feet from the mirror. What is the height of the streetlight to the nearest tenth of a foot?
4612
5.4x
37a WP: Similar s and proportions