Transcript Lesson #1 - Radical Tutor

Lesson #1
Points, Lines, Planes, and Circles
Lesson Goals
• Students will be able to define a point, a line,
a line segment, a ray, and a plane
• SWBAT define collinear points and concurrent
lines
• SWBAT recognize collinear points and
concurrent lines
• SWBAT define a circle
Do Now
• Draw a dot on a piece of paper.
• Discuss with your neighbor what shape you
might get if you drew a bunch of other dots all
the same distance from your original dot.
• Is there another shape? Hint: think “outside
the page.”
Points and lines
• Point
• Line Segment
• Ray
• Line
Planes
• Describe a plane in your own words and point
out at least one plane in this room to your
neighbor
Collinearity and Concurrency
• When three or more points…
• When three or more lines…
• Why three or more?
Conclusion and Homework
• Pg. 4 # 1-4
• Pg. 6 # 1-4
• Purchase compass, straightedge, textbook,
and graph paper notebook from the school
store
Lesson #2
The Five Axioms of Geometry
Lesson Goals
• SWBAT copy a line segment, using only a
compass and straightedge
• SWBAT name and understand Euclid’s five
axioms
Do Now: Pick a segment, any segment
• Draw a line segment on your page using your
straightedge
• Assuming you couldn’t measure distances with your
straightedge, how could you draw an identical line
segment with just a straightedge and a compass?
Axioms: When you can’t prove it, just
say it
• Euclid (b. ~300 BC)
• Wrote Elements, which is sort of a much more
impressive version of Introduction to
Geometry
• Started with five basic axioms
• What’s an axiom?
Euclid apparently
looked like Santa
No?
Five Axioms
• Any two points can be connected by a straight line
segment.
• Any line segment can be extended forever in both
directions, forming a line.
• Given any line segment, we can draw a circle with the
segment as a radius and one of the segment’s
endpoints as the center.
• All right angles are congruent [the same measure].
• Given any straight line and a point not on the line,
there is exactly one straight line that passes through
the point and never meets the first line.
Any two points can be connected by a
straight line segment.
Any line segment can be extended
forever in both directions, forming a
line.
Given any line segment, we can draw a circle with the segment
as a radius and one of the segment’s endpoints as the center.
All right angles are congruent.
Given any straight line and a point not on the line, there is
exactly one straight line that passes through the point and never
meets the first line.
Angles, and how to measure them
• An angle is…
Make your own angle
• Draw two line segments on your page so that
they share a common point
• Someone come up here and do the same…
• What tools do you have to measure an angle?
Some properties of angles
• What if…
– We know <BAC and <CAD, but we want <BAD?
– We know <BAD and <BAC, but we want <CAD?
B
C
A
D
Acute, right, obtuse
• Acute is…
• Right is…
• Obtuse is…
Conclusion and HW
• Pg. 20 # 1,2
• [Start Construction Worksheet]
Lesson #3
Vertical Angles and Parallel Lines
Lesson Goals
• SWBAT define a vertical angle, corresponding
angles, alternate interior angles, alternate
exterior angles, same-side interior angles, and
same-side exterior angles
• SWBAT find the measure of these angles when
given parallel lines and a transversal
Parallel Parking
• Draw a line segment in your notebook (this can
be freehand)
• To “parallel park” your line segment, it needs to
move its own length in the same direction, twice
its length at a 45 degree angle to your original
line segment, and then its length in a direction
parallel to your original line. Give your car
directions from its original location with your
protractor and construct this series of line
segments in your notebook.
Parallel Line Notation
Transversals
Why does a triangle have 180
degrees?
HW
• Pg. 30 # 1, 2, 3, 4, 5
• Pg. 46 # 28, 31, 32, 33, 34
Lesson #4
Triangles
Lesson Goals
• SWBAT find the missing angle in a triangle
given the measures of the other two angles
• SWBAT find missing angles in a triangle given
an exterior angle and vice versa
Do Now
• Anyone complete the proof from yesterday?
Triangle Facts
• Triangles have _______ sides
• The measures of the angles in a triangle sum
to _________
• There are different types of triangles: whether
a triangle is _________, ___________, or
____________ tells you the biggest angle
• Special triangles you should know are
__________, ___________, and _________
Finding the angles of a triangle
• Using a protractor, you find that two of the
angles in a triangle measure 30°
– Classify this triangle two different ways
– Find the third angle of this triangle
What if we only know how the angles
relate to one another?
• The triangular base of a see-saw is to be
constructed such that one of its base angles is
twice the degree measure of the other, and
the third angle is 20° less than three times the
smallest base angle. Find the exact measures
of all three angles.
Exterior Angles
A
B
C
D
If m<A=26°, and m<B=62°, find the m<ACD.
What do you notice? Will this always be true?
HW
• Study for vocabulary quiz tomorrow
• Pg. 35 # 1, 2, 3, 4, 5
• Pg. 39 # 1, 2, 3, 4
Lesson #5
Parallel Lines Revisited
Lesson Goals
• Students will understand what the converse of
a statement is
• Students will prove the converse of the
statement “If lines are parallel, then
corresponding angles are congruent”
Do Now
• In the diagram below, what can we say about
lines k and m? Why?
k
m
Statement converses
• If this, then that.
• PQ
• If that, then this.
• QP
• Are these logically equivalent? How do you
know?
Proof by contradiction
k
m
HW
• Pg. 46 # 35-46
Lesson #6
Congruent Triangles
Lesson Goals
• SWBAT define congruence
• SWBAT show triangles are congruent using
SSS, SAS, ASA, and AAS
• SWBAT show that ASS does not imply triangles
are congruent
• SWBAT use a CPCTC argument to make an
argument about a congruent triangle
Do Now
• Triangle worksheet
SIDE SIDE SIDE CONGRUENCE (SSS)
I) Construct (with a straightedge and compass) a triangle with the following 3 sides.
Is it possible to construct a different triangle (with different interior angles) that has the same three side
lengths? Try to construct one.
CONCLUSION: IF TWO TRIANGLES HAVE 3 SIDES OF EQUAL LENGTH THEN THE TRIANGLES ARE
___________________
II. SIDE – ANGLE – SIDE CONGRUENCE (SAS)
Copy the angle shown below such the length of the two sides of the angle are equal in length to the two
segments shown below. Construct a triangle by completing the third side.
You have just constructed a triangle with two sides and an “included” angle. Can you construct another,
different triangle with the same included angle and 2 sides?
CONCLUSION: IF ONE ANGLE AND THE LENGTH OF ITS TWO ADJACENT SIDES OF ONE TRIANGLE ARE
EQUAL TO AN ANGLE AND THE LENGTH OF ITS TWO ADJACENT SIDES OF A SECOND TRIANGLE, THEN
THE TRIANGLES ARE ___________________
III. ANGLE – SIDE – ANGLE CONGRUENCE
Copy the following segment below. At both ends of the segment, construct angles that are equal in
measure to the two given angles. Extend the legs of the two angles until they intersect to construct a
triangle.
Is it possible to make a different triangle with the two given angles and the included side?
CONCLUSION: IF TWO ANGLES AND THE SIDE INCLUDED BETWEEN THEM IN ONE TRIANGLE ARE EQUAL
TO TWO ANGLES AND THE SIDE INCLUDED BETWEEN THEM ON A SECOND TRIANGLE, THEN THE TWO
TRIANGLES ARE _____________________________
IV. ANGLE – SIDE – SIDE CONGRUENCE
Copy the angle given below, extending the legs of the angle very lightly on your paper. Copy the longest
of the two segments below such that the segment is the length of one leg of the constructed angle.
Connect the second, shorter segment below so that one end of the segment connects to the end of the
first segment and the other end intersects with the remaining leg of the copied angle to construct a
triangle.
Is there any other way to construct a different triangle in the same “angle – adjacent side – opposite
side configuration?
CONCLUSION: IF ONE ANGLE AND TWO SEGMENTS (NOT THE INCLUDED ANGLE) OF ONE TRIANGLE
ARE EQUAL TO ONE ANGLE AND TWO SEGMENTS OF A SECOND TRIANGLE THEN
______________________________________________
Lesson #7
Congruent Triangles
Lesson Goals
• SWBAT show triangles are congruent using
SSS, SAS, ASA, and AAS
• SWBAT show that ASS does not imply triangles
are congruent
• SWBAT use a CPCTC argument to make an
argument about a congruent triangle
Do Now
Find <BAD. How do you know? (Be careful with your
reasoning)
•
8
A
B
3
3
32°
C
8
D
Proof
• Prove that if a radius of a circle bisects a chord
of the circle that is not a diameter, then the
radius must be perpendicular to the chord
• Hint: Use the triangle properties you now
know and love.
SAS
• Prove that AB || CD
B
A
6
5
5
D
6
C
Let’s fly a kite!
• Prove that DB is perpendicular to AC
HW
• Pg. 54 #1-4
• Pg. 58 #1-4
Lesson #8
Congruent Triangles
Lesson Goals
• SWBAT show triangles are congruent using
SSS, SAS, ASA, and AAS
• SWBAT show that ASS does not imply triangles
are congruent
• SWBAT use a CPCTC argument to make an
argument about a congruent triangle
Do Now: Prove it
Given: AD=BC, AD || BC, E and F are on AC, <ADE=<CBF
Prove: AB || CD
AND
Prove: DF=EB
A
B
E
F
D
C
Which of the following triangles are
congruent?
16”
16”
28°
28°
55°
13.2”
16”
28°
97°
13.2”
16”
97°
HW
• Pg. 63 # 1 - 4
• Pg. 75 #24, 26, 27, 28, 29, 30
Lesson #9
Isosceles and Equilateral Triangles
Lesson Goals
• SWBAT define isosceles and equilateral
triangles
• SWBAT to utilize properties of isosceles and
equilateral triangles to solve problems
• SWBAT to utilize properties of isosceles and
equilateral triangles for geometric proofs
Do Now: Equilateral Triangles
• An equilateral triangle has three equal sides
• Prove that all its angles are equal too.
Definitions
• An isosceles triangle is a triangle with two
equal sides (the legs) and one unequal side
(the base)
• An isosceles triangle has base angles and a
vertex angle
• An equilateral triangle has three equal sides
Problem solving with types of triangles
• XY=XZ=14 and <X=42°
• Find the other two angles.
X
Y
Z
Proofs with types of triangles
• Given: m<ABD=m<ACD, m<BAD=1/2m<ABD,
m<BAD=m<CAD
• Prove: ABC is an equilateral triangle
A
B
C
D
HW
• Pg. 71 #1-7
Lesson #10
Area and Perimeter
Lesson Goals
•
•
•
•
SWBAT define area
SWBAT define perimeter
SWBAT find the area of a grid-based shape
SWBAT find the perimeter of a polygon
Guard the perimeter! Search the area!
• What do these statements mean?
• Come up with a definition for perimeter and
write it in your notes.
• Come up with a definition for area and write it
in your notes.
Find the perimeter
14
18√2
Find the area
Word problems with perimeter
• The length of each leg of an isosceles triangle
is three times the length of the base of the
triangle. The perimeter of the triangle is 91
cm. What is the length of the base of the
triangle?
HW
• Pg. 83 #1-5
• Pg. 88 #1-6
• STUDY
Lesson #11
Area
Lesson Goals
• SWBAT find the area of a rectangle, square,
and triangle
Do Now
• Mr. P would like to paint his classroom blue,
his happy color. Unfortunately he does not
know the area he needs to paint.
• Can you determine this, approximately and in
square feet, using the classroom rulers?
Finding areas of various rectangles
What is a general formula for finding the area of a rectangle?
Area of a right triangle
What is a general formula for finding the area of a triangle? How does it relate to the
formula for the area of a rectangle and why?
Problems with area
• The length of one side of a rectangle is 4 less
than 3 times an adjacent side. The perimeter
of the rectangle is 64. Find the area of the
rectangle.
Areas of non-right triangles
• Find the area of triangle DEF.
HW
• Pg. 88 #1-6
– 6 is hard.
Lesson #12
Same Base, Same Altitude
Lesson Goals
• Students will understand and solve problems
using the same base principle
• Students will understand and solve problems
using the same altitude principle
Do Now
Determine [ABC]/[ACD] and [ABC]/[ABD].
Same altitude property
• If two triangles share an altitude, then the
ratio of their areas is the ratio of the bases to
which that altitude is drawn
• This is particularly useful for problems in
which two triangles have bases along the
same line
Find the ratio
• Determine the ratio of [ABC]/[ABD]
C
D
A
B
Same base property
• If two triangles share a base, then the ratio of
their areas is the ratio of the altitudes to that
base
HW
• Pg. 91 # 1 – 5
• Pg. 93 # 13, 14, 15, 16, 17
Lesson #13
Triangle Similarity
Lesson Goals
• SWBAT define the term “similarity”
• SWBAT determine AA similarity in a triangle
• SWBAT solve for sides of a triangle based on
AA similarity
Do Now
• The shapes below are identical, except one is a “blown-up”
version
• Discuss with a partner: what do you think you can say
about the angles and the sides in this shape?
Similar Shapes
• We call two figures similar if one is simply a
blown-up, and possibly rotated and/or
flipped, version of the other
• Similar figures will have IDENTICAL ANGLES
and their SIDES WILL BE IN THE SAME
PROPORTION to one another
AA Similarity
• Two triangles are similar if they have two
identical angles
• Why not 3 identical angles?
Can you solve for the value of x?
12
x
5
10
Can you solve for x and y?
3
y
5
4
x
Another note on this diagram
x / =y /
w
z
Why?
z
y
x
w
HW
• Pg. 101 # 1, 2
• Pg. 108 # 1, 2
• Extra Credit Offering (For 3 additional points
on your last examination IF you scored below
an 85 or 1 additional point IF you scored over
an 85): Pg. 96 # 32, 34, 35, 38, 39
Lesson #14
SAS Similarity
Lesson Goals
• SWBAT find similar triangles using SAS
• SWBAT use SAS similarity to solve for sides of
a triangle
Do Now
• Solve for x.
4
7
9
x
What can you say about the triangles
below?
10
5
2.5
5
SAS Similarity
• If two sides in one triangle are in the same
ratio as two sides in another triangle, and the
angles between these sides are equal, then
the triangles are similar
A hard example!
• Given AC=4, CD=5, and AB=6 as in the
diagram, find BC if the perimeter of BCD is 20
B
6
A
D
4
C
5
HW
• Pg. 112 # 1-4
Lesson #15
SSS Similarity
Lesson Goals
• SWBAT find similar triangles using SSS
• SWBAT use SSS similarity to solve for sides of a
triangle
Do Now: Conjectures?
• What do you think we can say about the two
triangles below? Why? (Be specific)
4
14
7
5
10
8
SSS Similarity
• SSS similarity tells us that if each side of one
triangle is the same constant multiple of the
corresponding side of another triangle, then
the triangles are similar
• Corollary: SSS similarity tells us that their
corresponding angles are equal
Using SSS Similarity
• Given the side lengths shown in the diagram,
prove that AE || BC and AB || DE
A
5
B
4
12
6
C
4
E
10
D
HW
• Pg. 114 #1
Lesson #16
Using Similarity in Problems
Lesson Goals
• SWBAT use their knowledge of AA, SAS, and
SSS similarity to solve problems
Do Now
• In the diagram, DE || BC, and the segments
have the lengths shown in the diagram. Find
x, y, and z
A
z
45
E
D
27
36
64
B
60
y
x
C
Practice Problem #1
(a) Use similar triangles to find ratios of segments that equal EF/AB
(b) Use similar triangles to find ratios of segments that equal EF/DC
(c) Use one ratio from each of the first two parts and add them to get an
equation you can solve for EF
D
A
E
9
12
B
C
F
40
Areas and Similarity
• ABC ~ XYZ, AB/XY=4, and [ABC]=64. Draw this.
• Let c be the altitude of ABC to AB and let z be
the altitude of XYZ to XY. Draw this.
• What is c/z?
• Find [XYZ].
• Can you make a general statement about the
area of similar triangles?
Area and Similarity
• If two triangles are similar such that the sides
of the larger triangle are k times the size of
the smaller, then the area of the larger
triangle is k2 times that of the smaller!
Proof with Similarity
• In the diagram, PX is the altitude from right angle
QPR of right triangle PQR as shown. Show that
PX2=(QX)(RX), PR2=(RX)(RQ), and PQ2=(QX)(QR).
• How does the transitive property come into play
here?
P
Q
X
R
HW
• Pg. 115 # 16, 18
• Pg. 120 # 1-4
Lesson #17
Lesson Goals
• SWBAT define the legs and hypotenuse of a
right triangle
• SWBAT to prove the Pythagorean Theorem
(just one of the many proofs)
• SWBAT use the PT to find the sides of a right
triangle
Do Now
• Prove that a2=cd
• Prove that b2=ce
• Use the two statements above to show that
a2+b2=c2
a
b
d
e
c
Key Vocabulary
• Pythagorean Theorem: a2+b2=c2, where a and
b are the legs of a right triangle, and c is the
hypotenuse of the same right triangle
Find the missing side
3
4
Find the missing side
6
10
Find the missing side
3
5
4
9
x
HW
• Pg. 139 # 1, 2, 4, 5
Lesson #18
Two Special Right Triangles
Lesson Goals
• SWBAT find the side lengths of a 30-60-90
triangle, given one side
• SWBAT find the side lengths of a 45-45-90
triangle, given one side
Do Now: 45-45-90
• Using Pythagorean Theorem, show that side
AB and BC must both be 1
A
√2
B
C
Side Note
• The √2 is called “irrational”
• The Pythagorean who determined that it was
irrational was killed
Find the length of x in each of the
following: can you write a rule?
2
1
3
4
30-60-90
• A 30-60-90 triangle will have sides in the ratio
1:√3:2
• Here’s why…
Proof
Finding the sides of a 30-60-90 triangle
1
8
y
y
x
x
HW
• Pg. 146 # 1, 2
Lesson #19
Pythagorean Triples
Lesson Goals
• SWBAT recognize Pythagorean triples
• SWBAT generate an infinite number of
Pythagorean triples based on a given {a,b,c}
triple
• SWBAT generate an infinite number of
Pythagorean triples using even numbers
Pythagorean Triples
• A Pythagorean triple is a set of three whole
numbers (integers greater than 0) that satisfy
the Pythagorean Theorem
Pythagorean Triple Contest!
• Split into groups of 3 and write as many
Pythagorean triples as you can in 5 minutes
• The winners shall be held up in the glory of
the SUNSHINE CORNER and receive an
additional 10 points on their next homework
Prove it
• Given {a,b,c} is a set of Pythagorean triples,
prove that {na,nb,nc} is a set of Pythagorean
triples for any whole number n
How to generate a massive amount of
triples
•
•
•
•
•
•
•
•
•
Take any even number and call it a
Divide by 2
Square it
Call this number z
Subtract 1 from z
Call this number b
Add 1 to z
Call this number c
{a,b,c} is a Pythagorean triple
Big Pythagorean triples to know
•
•
•
•
{3,4,5}  The Granddaddy of them all
{5,12,13}
{7,24,25}
{8,15,17}
HW
• Finish problems from last lesson
• Pg. 151 # 1, 3, 4
Lesson #20
Congruence and Similarity Revisited
(in the context of right triangles and
the PT)
Lesson Goals
• SWBAT prove two right triangles congruent
given two sides
• SWBAT prove two right triangles similar given
two sides
Do Now
• Prove that the two right triangles below are
congruent. What can you say if you are given
two right triangles with identical hypotenuses
and one identical leg?
15
12
12
15
Hypotenuse-Leg Congruence
• HL congruence states that if the hypotenuse
and a leg of one right triangle equal those of
another, then the triangles are congruent.
• Note you don’t need leg-leg congruence,
because you already have it by SAS.
Hypotenuse-Leg Similarity
• Prove the two triangles below are similar:
15
12
HL Similarity
• HL Similarity states that if the hypotenuse and
a leg of one right triangle are in the same ratio
as the hypotenuse and leg of another right
triangle, then the two triangles are similar
If a radius of a circle bisects a chord of
a circle…
• The center of a circle is 4 units away from a
chord PQ of the circle. If PQ=12, what is the
radius of the circle?
HW
• Pg. 155 #1, 2, 4
Lesson #21
Heron’s Formula
Lesson Goals
• SWBAT state Heron’s formula
• SWBAT apply Heron’s formula to find the area
of a triangle, given three side lengths
Do Now
• Find the area of the two triangles below:
7
7
9
9
7
9
Heron’s Formula
• [Proof]
• You will not need to prove Heron’s (it’s quite a
lot of algebra), but you will need to be able to
apply it
• Heron’s formula states that given three sides
of a triangle, {a, b, c}, the area of the triangle
is √(s(s-a)(s-b)(s-c)), where s=(a+b+c)/2
Applying Heron’s Formula
• Use Heron’s Formula to find the area of the
triangles below
7
7
11
9
7
7
HW
• Pg. 160 #1, 2
• Spend 10 minutes (then stop if you are hitting
a wall) going through the Heron’s formula
proof, just for your own edification…
Lesson #22
Perpendicular Bisectors of a Triangle
Do Now
• From what two points must every point on k be equidistant?
• From what two points must every point on m be equidistant?
Lesson Goals
• Students will be able to define concurrent,
circumcenter, circumradius, circumcircle
• SWBAT evaluate the circumradius of a triangle
for all triangles and the special case of the
right triangle
Perpendicular bisectors of the sides of
a triangle are CONCURRENT
• Lines are concurrent if they all meet at a single point
• The point at which the perpendicular bisectors of a
triangle meet is called the circumcenter
• The circle centered at the circumcenter that passes
through the vertices of the original triangle is called
the circumcircle, which is circumscribed about the
triangle
• The circumradius is the radius of this circle
Circumcenter of a right triangle
• Construct at least TWO right triangles in your
books using a protractor and a straightedge
• Create perpendicular bisectors of all three
sides for your triangles
• Where is the circumcenter in all of your
triangles?
Circumcenter of a right triangle
• The circumcenter of a right triangle is the
midpoint of the hypotenuse
• The circumradius is ½ the length of the
hypotenuse
• Therefore, the hypotenuse is the diameter of
the circumcircle
How many points define a circle?
• 1, 2, 3, more?
Find the circumradius
• Find the circumradius of an equilateral
triangle with side length 6
HW
• Pg. 177 #2, 3, 4
Lesson #23
Angle Bisectors of a Triangle
Lesson Goals
• SWBAT define the incenter, inradius, and
incircle of a triangle
• SWBAT derive and understand the angle
bisector theorem
• SWBAT find the area of a triangle given its
inradius and its side lengths
Do Now
• Construct a triangle and bisect two of its
angles
• What can you say about the bisector of the
third angle?
Key Vocabulary
• The angle bisectors of a triangle are
concurrent at a point called the incenter
• The common distance from the incenter to
the sides of the triangle is called the inradius
• The circle inscribed in the triangle is called the
incircle
– NOTE: Each triangle has only one incircle, whose
center is the intersection of the angle bisectors of
a triangle
The Angle Bisector Theorem
• Given: Triangle ABC with BE its angle bisector
• Then: AB/AE=CB/CE
B
A
C
E
How to use Angle Bisector Theorem
• Find AC in the diagram
A
12
C
7
B
6
D
Results from bisecting an angle
• Distance from all three sides is equal at the
incenter (note this was not the case with
perpendicular bisectors)
• Therefore the incircle is tangent to each side
of the triangle at just one point and is
inscribed within the triangle
Finding the Area of a Triangle from its
inradius
• Recall that all the perpendicular lines drawn to
the sides from the inradius are equal in length
• Can you write a formula for the area of the
triangle given an inradius of length r and side
lengths of a, b, and c?
Finding the Area of a Triangle from its
inradius
• The area of a triangle equals its inradius times
its semiperimeter (s=(a+b+c)/2)
• Example: Find the radius of a circle that is
tangent to all three sides of triangle ABC,
given that the sides of ABC have lengths 7, 24,
and 25
HW
• Pg. 182 #1, 2, 3, 5, 7
Lesson #24
Medians
Lesson Goals
• SWBAT define median, centroid, and medial
triangle
• SWBAT show that medians divide the triangle
into 6 triangles of equal area
• SWBAT show that the centroid cuts each
median into a 2:1 ratio
• SWBAT prove the midline theorem
New Vocabulary
• A median of a triangle is a segment from a
vertex to the midpoint of the opposite side
• The medians of a triangle are concurrent at a
point called the centroid of the triangle
Do Now
• Show that the medians of triangle ABC cut
the triangle into six triangles of equal area
C
A
B
Medians and ratios
• Show that the centroid of any triangle cuts each of the triangle’s medians
into a 2:1 ratio, with the longer portion being the segment from the
centroid to the vertex
C
A
B
The Medial Triangle
• In ABC below, DEF is referred to as the medial
triangle
C
D
E
A
F
B
Prove the four smaller triangles
below are congruent
C
D
E
A
F
B
The Midline Theorem
• DEF~ABC, DEF=FBD=AFE=EDC
• EF/BC=DE/AB=DF/AC=1/2
• DF||AC, EF||BC, DE||AB
C
D
E
A
F
B
HW
• Pg. 187 #1, 2, 3
Lesson #25
Altitudes
Lesson Goals
• SWBAT define orthocenter
• SWBAT prove that the lines containing the
altitudes of any triangle are concurrent
• SWBAT solve problems involving the
properties of altitudes
Do Now
• Where do you think the altitudes of a right
triangle intersect? (Don’t prove this; just use a
few examples)
Prove: Altitudes are concurrent
•
•
•
•
•
•
This is legitimately very hard…
Draw a line parallel to BC through A, parallel to AB through C, and parallel to AC through B
– The intersections of these lines form another triangle, which we’ll call JKL
Prove CAK=ACB
Show that A, B, and C are the midpoints of KL, JL, and JK, respectively
Describe the relationship of AD, BE, and CF to JKL
What does this imply?
C
D
E
A
F
B
Vocabulary
• The altitudes of any triangle are concurrent at
a point called the orthocenter
Example involving altitudes
• Altitudes QZ and XP of XYZ intersect at N.
Given that <YXZ=70° and <XZY=45°, find:
– m<ZXP
– m<XZQ
– m<YXP
Point of interest with orthocenters
• The altitudes of ABC meet at point H. At what
point do the altitudes of ABH meet? How
about ACH?
C
D
E
H
A
F
B
HW
• Pg. 192 #1, 2, 4 (this is a proof), 5 (this is a
proof also)
Lesson #26
Introduction to Quadrilaterals
Lesson Goals
• SWBAT define a quadrilateral by its sides,
vertices, and angle measures
• SWBAT find the measures of angles of a
quadrilateral
Do Now
• Write a full proof demonstrating how many
degrees are in the sum of the angles of a
convex quadrilateral (below).
Two types of quadrilaterals
• Convex
• Concave
>180 degrees!
Other major types
• Think/pair/share: Name some other
quadrilaterals that you know.
Angles in a quadrilateral
• Prove that any convex quadrilateral has angles
of total measure 360 degrees
Finding angle measures in a
quadrilateral
• A quadrilateral has angles of measure x,
3x+20, 2x-20, and 6x+12.
• Find all the angles in the quadrilateral and
sketch what it might look like.
• Is the quadrilateral concave or convex?
HW
• Pg. 208 #1, 2, 3
Lesson #27
Trapezoids
Lesson Goals
• SWBAT define a trapezoid and the special
types of trapezoids
• SWBAT find angle measures in a trapezoid
• SWBAT find the area of a trapezoid
Finding the area of a trapezoid
• Below is a trapezoid, a quadrilateral with
(only) two parallel sides. Using what you
know about the area of rectangles and
triangles, find the area of the trapezoid
6
8
12
Area of a trapezoid
• If x and y are the lengths of the two bases and
h is the height of a trapezoid ABCD,
[ABCD]=(x+y/2)(h)=(the average of the base
lengths)(height)
Finding the area of a trapezoid
• Find the area of the below trapezoid
13
7
4
Angles in a trapezoid
• Find the base angles in the trapezoid below:
106°
Trapezoids and Parallel Lines
• Most problems with trapezoids can be
reduced to the facts about parallel lines and
similar triangles we learned at the beginning
of the year
Special Type of Trapezoids
• Isosceles trapezoids have:
– Two equal-length legs
– Congruent base angles
– Equal-length diagonals
– ANY OF THESE DEFINES AN ISOSCELES
TRAPEZOID!
HW
• Pg. 214 #1-4
Lesson #28
Parallelograms
Lesson Goals
• SWBAT define a parallelogram
• SWBAT find that a shape is a parallelogram
based on its diagonals
Do Now
• A parallelogram is a quadrilateral made up of
two pairs of parallel sides
• Find x, y, and <C in the parallelogram below
x+y
30°
3x
Prove: AE=CE
• Given: ABCD is a parallelogram
A
B
E
D
C
The diagonals of a parallelogram
• The diagonals of a parallelogram bisect one
another, as you just proved
Area of a parallelogram
• Find the area of the parallelogram below
• Hint: remember how we proved the area of a
trapezoid
14
11
15
Area of a parallelogram
• Easy!
• A=bh
• Just like a rectangle (intuitive geometric way
of showing this?)
HW
• Pg. 218 #1, 2, 3, 5