Download All you ever wanted to know about Triangles

Unit 8 –Triangles
•This unit continues with triangles.
•It defines midsegments, bisectors,
altitudes, medians, centroids, points of
concurrency, incenters, inscribed
circles, circumscribed circles,
midpoints and perpendicular bisectors.
• It also proves triangles are congruent
by SSS, SAS, ASA, AAS, HL, and
CPCTC.
Standards
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SPI’s taught in Unit 8:
SPI 3108.1.1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and space.
SPI 3108.4.11 Use basic theorems about similar and congruent triangles to solve problems.
SPI 3108.4.12 Solve problems involving congruence, similarity, proportional reasoning and/or scale factor of two similar figures
or solids.
CLE (Course Level Expectations) found in Unit 8:
CLE 3108.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including testing cases, estimation, and
then checking induced errors and the reasonableness of the solution.
CLE 3108.1.4 Move flexibly between multiple representations (contextual, physical written, verbal, iconic/pictorial, graphical,
tabular, and symbolic), to solve problems, to model mathematical ideas, and to communicate solution strategies.
CLE 3108.1.5 Recognize and use mathematical ideas and processes that arise in different settings, with an emphasis on
formulating a problem in mathematical terms, interpreting the solutions, mathematical ideas, and communication of solution
strategies.
CLE 3108.1.7 Use technologies appropriately to develop understanding of abstract mathematical ideas, to facilitate problem
solving, and to produce accurate and reliable models.
CLE 3108.4.8 Establish processes for determining congruence and similarity of figures, especially as related to scale factor,
contextual applications, and transformations.
CFU (Checks for Understanding) applied to Unit 8:
3108.1.5 Use technology, hands-on activities, and manipulatives to develop the language and the concepts of geometry,
including specialized vocabulary (e.g. graphing calculators, interactive geometry software such as Geometer’s Sketchpad and
Cabri, algebra tiles, pattern blocks, tessellation tiles, MIRAs, mirrors, spinners, geoboards, conic section models, volume
demonstration kits, Polyhedrons, measurement tools, compasses, PentaBlocks, pentominoes, cubes, tangrams).
3108.1.7 Recognize the capabilities and the limitations of calculators and computers in solving problems.
3108.4.35 Prove that two triangles are congruent by applying the SSS, SAS, ASA, AAS, and HL congruence statements.
3108.4.36 Use several methods, including AA, SSS, and SAS, to prove that two triangles are similar.
3108.4.38 Use the principle that corresponding parts of congruent triangles are congruent to solve problems.
3108.4.41 Use inscribed and circumscribed polygons to solve problems concerning segment length and angle measures.
A basic review before we
move on…
• Triangle Sum theorem: The sum of the
measure of the angles of a triangle is….
– m of angle A plus angle B plus angle C = 180
• Triangles classified by sides:
– Equilateral (all sides equal)
– Isosceles (at least 2 sides equal)
– Scalene (no sides equal)
• Triangles classified by angles:
–
–
–
–
Equiangular (all angles equal)
Acute (all angles acute) (less than 90 degrees)
Right (one right angle) (equal to 90 degrees)
Obtuse (one obtuse angle) (greater than 90 degrees)
Still reviewing…
• The measure of each exterior angle of a
triangle equals the sum of the measures
of its two remote interior angles.
• m of angle 1 = m of angle 2 and 3
If Angle 3 is 90
degrees, what is
the measure of
angle 2?
What is the measure
of Angle 1?
2
3
25o
1
Mid-segments of Triangles
• Picture this triangle:
• Now imagine if you folded the top over, so
that it touched the bottom corner. To make
this fold, we connect the half-way points of
each side of the triangle
• By appearances, what
conclusions can we draw
about this picture?
Midsegment
line
X
X
• The mid-segment line that we drew is
parallel to the bottom line of the triangle
• The mid-segment line is half as long as the
bottom line of the triangle.
X
Triangle Mid-segment Theorem
• Therefore we can create a theorem to capture these
ideas:
• If a segment joins the midpoints of two sides of a
triangle, then the segment is parallel to the third side,
and is half its length
• Converse thinking comes in handy here as well -> if
it’s not connecting two midpoints, then its not
parallel, and it’s not half as long…
• …and, if a segment connects a midpoint, it’s parallel
to a side…
• Therefore, three segments connecting midpoints
would create three parallel sides, and you would
have a triangle which is half as big…whew…
Example
•
•
•
•
Assume r and s and t are midpoints
Find the length of segment rs
Find the length of segment st
Find the length of segment pq
40
p
r
o
s
t
q
100
•Segment rs is parallel to segment oq, therefore rs = 1/2 segment oq, or
1/2 * 100 = 50
•Segment st is parallel to segment op, therefore st = 1/2 segment op, or
1/2 * 60 = 30
•Segment pq is parallel to segment rt, therefore pq = twice segment rt, or
2 * 40 = 80
Assignment
• Page 288-289 7-26
• Worksheet 5-1
Unit 8 Quiz 1
1.
2.
3.
4.
5.
6.
7.
What is the length of segment AC?
What is the length of segment DF?
What is the length of segment FG?
Is segment AD congruent to segment DG?
Is segment BG congruent to segment DF?
Name a segment parallel to segment DG
Name all segments congruent to segment
AF
8. When classifying points, what do we call
points D,F, and G?
9. When classifying segments, what do we
call segment DF, Segment DG, and
Segment FG?
10. What type of triangle is Triangle ABC?
(right/obtuse/acute)?
A
4
F
D
4
14
B
6
G
6
C
Bisectors in Triangles
• This is really a review, but hey…
• Theorem - If a point is on the perpendicular
bisector of a segment, then it is equidistant
from the endpoints of the segment (that’s why
it’s called a bisector, right?)
• Converse: If a point is equidistant from the
endpoints of a segment, then it is on the
perpendicular bisector of the segment
Angle Bisector Theorem
• Theorem - If a point is on the bisector of
an angle, then the point is equidistant
from the sides of the angle
• Converse - If a point in the interior of an
angle is equidistant from the sides of
the angle, then the point is on the angle
bisector
Examples
• Now we can apply this to
triangles and such…
• We can see that point f is on
the bisector of angle abe
• Therefore we can draw a
line from f to b and f to d,
and those lines are equal
• Solve for x, then determine
the length of fd and fb
a
b
5x
f
c
2x+24
d
e
Assignment
• P. 296-297 7,8,12-20
• Worksheet 5-2
Make a Quiz
• Task: You must make a 10 question quiz
• This quiz must have questions on:
1.
2.
3.
Mid-segments
Angle Bisectors
Perpendicular Bisectors
• You must work with only ONE partner
• You must have proper pictures, labels etc.
• You must make an answer key, and check all your
questions to make sure they work
• You will give this test to someone else in this class
• You will grade their answers
• You have 20 minutes to make this test
Concurrent Lines, Medians,
and Altitudes
• Question: What do you think the result
would be if you bisected all the angles
in a triangle?
x
x
x
• They all meet in one point
• When 3 or more lines intersect in one point, they are considered
“concurrent.”
• Therefore the answer is “The lines intersect in a point of
concurrency.”
•Theorem - The bisectors of the angles of a triangle are concurrent at
a point equidistant from the sides.
Perpendicular Bisectors
• Question: What do you think happens if
you draw perpendicular bisectors for
the sides of a triangle?
x
• Answer: they are
concurrent, and…
x
x
• The perpendicular bisectors of the sides of a
triangle are concurrent at a point equidistant
from the vertices (corners)
And how do I find the centers
of Circles (a life question)?
• How do I find the center of a circle
drawn around a triangle? (we call that
circumscribed)
• Draw perpendicular
bisectors (the 3rd one isn’t
required here since the other
two intersect exactly at the
Remember, we want
rd
midpoint of the 3 side –
to be the exact same
because it’s a right triangle) distance from all 3
corners
The Circumcenter
• A circumscribed circle is one that
goes AROUND the triangle
• A circumcenter is the center point
of this triangle
• In an acute triangle, the
circumcenter is inside the triangle
• In a right triangle the circumcenter
is on one side of the triangle
• In an obtuse triangle the
circumcenter is on the outside of
the triangle
And the inside?
• What if I draw a circle inside the
triangle? (We call that inscribed)
• This time we use angle bisectors…
• And again, we don’t
need the 3rd line (we just
need 2 lines to intersect),
but it looks cool.
• We call the point of
concurrency of the angle
bisectors the Incenter of
the triangle.
Medians
• A median of a triangle is a segment whose end points
are 1) a vertex (or angle), and 2) the midpoint of the
opposite side
• So it probably isn’t an angle bisector (although it
could be).
• Theorem 5:8 - The medians of a triangle are
concurrent at a point that is two thirds the distance
from each vertex (corner) to the midpoint on the
opposite side.
2/3
1/3
Medians
• In a triangle, the point of concurrency of
the medians is called the centroid. This
is also considered the center of gravity
of a triangle, because it is the point
where a triangular shape will balance
(like on the point of a pencil)
Centroid
Altitudes of Triangles
• An altitude of a triangle is the perpendicular
segment from a vertex (corner) to the line
containing the opposite side
• Unlike angle bisectors and medians, an
altitude of a triangle can be a side of a
triangle, inside it, or it may lie outside the
triangle
Acute (inside)
Right (is the side)
Obtuse (outside)
Assignment
• Page 312 8-13,
• Page 313 17-20,24-27
• Worksheet 5-3
Unit 8 Quiz 2
1.
2.
3.
4.
A mid-segment is twice as long as the side it parallels (true/false)
An altitude is “how high” a triangle is (true/false)
When three points intersect, it is called a point of __________ (fill in blank)?
Where the three medians of a triangle intersect it is called the _________ (fill
in the blank)?
5. The intersection of a) Angle Bisectors, or B) Perpendicular Bisectors is used
to find the center point of the circle that goes outside the triangle (A or B)?
6. The intersection of a) Angle Bisectors, or B) Perpendicular Bisectors is used
to find the center point of the circle that goes inside the triangle (A or B)?
7. If a circle is inside the triangle, it is called _______________ ? (Starts with I)
8. If a circle is outside the triangle, it is called ______________ ? (Starts with C)
9. A line that goes from the corner, to the opposite midpoint is called the
_______________ ? (fill in blank)
10. A median is broken into two lengths, the long length before the median
intersects with other medians, and the short length after the intersection of
medians. If the length of the median is 9 inches, how long is the long
section, and how long is the short section?
Comparing Sides of a Triangle
• If two sides of a triangle are not equal,
(congruent) then the largest angle lies
opposite the largest side
• If two angles of a triangle are not congruent,
then the longest side lies opposite the
largest angle
• The sum of the lengths of any two sides of
a triangle must be greater than the length
of the third side
• These all seem pretty common sense, eh?
Examples
Largest angle
Largest side
5
5
12
The sum of two sides are not
greater than 3rd side, therefore
they cannot connect
Example
• Algebra: A triangle has side lengths of 8 cm and 10
cm. Use inequalities to describe possible lengths for
the 3rd side.
• Let X = the length of the 3rd side
• Try all three combinations of sides:
• X + 8 > 10 or --> X > 2
• X + 10 > 8 or --> X > -2 (not a solution)
• 8 + 10 > X or --> X < 18
• Therefore, 2 < X < 18
• The shortcut (whoo hoo!) is this: “Big minus little,
Big plus little, put X in the middle…”
Example
• A triangle has side lengths of 3 and 12
inches. Describe the lengths possible for the
3rd side.
• X + 3 > 12, so X > 9
• X + 12 > 3, so X > -9
• 3 + 12 > X, so X < 15
• Therefore, 9 < X < 15
• Again, we can just subtract the 2 numbers (as
long as I get a positive answer), and add the 2
numbers, and put X in between them.
Assignment
• Page 329 9-32
• Worksheet 5-5
Wheel Line Irrigation
Unit 8 Quiz 3
• Farmers use wheel line irrigation systems to water fields where they don’t
get enough rain
• Some of these rotate around a pivot point in the middle of the field,
creating a perfect circle of watered plants
• When the wheel line gets to a corner where the circle won’t reach, a water
cannon –known as a gun- will shoot water into the corners.
• If a farmer has the triangular field shown below, how
does he know where to put the pivot point, so that he has
a perfectly inscribed circle?
.
Congruent Triangles
• Question: If you took 3 straws of
different lengths (or not, it doesn’t
matter), and made a triangle, and took 3
more straws that were exactly the same
length, and made another triangle,
would it be the same?
• Does it have to be the same, or could
you make more than one different
triangle?
Side-Side-Side (SSS)
Postulate
• If the three sides of a triangle are
congruent to three sides of another
triangle, then the two triangles are
congruent.
6
5
4
5
6
4
Side-Angle-Side (SAS)
Postulate
• If two sides and the included angle (the
one in-between the two sides) are
congruent to the two sides and included
angle of another triangle, then the
triangles are congruent
Assignment
• Page 231 11-14
• Page 232 24-26
• Worksheet 4-2
Unit 8 Quiz 4
•
•
•
•
Miguel wants to make a triangular shaped deck. The side of the
deck that runs along the house is called the ledger, and is 15
feet long. The second side of the deck angles out from the
house, and is 10 feet long.
Miguel is trying to figure out the third side of the deck. He
wants to make the third side with just one board.
Draw a picture (2 points)
Label the picture with all information (2 points)
1.
2.
•
•
How short can this board be and still make a triangle? (2 points)
How long can this board be and still make a triangle? (2 points)
Hint: Since he doesn’t know how long, Miguel calls the length
of the board “X”
Put your name on the paper (2 points)
Angle-Side-Angle Postulate
(ASA)
• If two angles and the included side (the
one between the two angles) of one
triangle are congruent to two angles and
the included side of another triangle,
then the triangles are congruent
Angle-Angle-Side (AAS)
Theorem
• If two angles, and a non-included side
of one triangle are congruent to two
angles and the corresponding nonincluded side of another triangle, then
the triangles are congruent
Assignment
•
•
•
•
Page 238 8-9
Page 239 16-18
Page 243 6-11
Worksheet 4-3
Using Congruent Triangles:
(CPCTC)
• We used SSS, SAS, ASA, and AAS to show
that triangles are congruent
• We could then draw conclusions about other
parts of the triangles, because by definition all
parts of congruent triangles are congruent.
• We will call this “corresponding parts of
congruent triangles are congruent” or CPCTC
Example
•
•
•
•
Believe it or not, this is the framework to an umbrella :)
Given: segment sl = segment sr
c
Given: angle 1 = angle 2
3 4
Prove angle 3
and angle 4 are
l
congruent
r
5
6
1 2
s
We can prove that triangle SIC is congruent to triangle SRC by SAS
Therefore, we can conclude that all sides and angles are congruent, by our
original definition of congruency
Therefore, angle 3 and angle 4 are congruent
This again proves that CPCTC
Hypotenuse Leg (HL)
Congruency in Right Triangles
• Theorem 4:6 -If the hypotenuse and a
leg of one right triangle are congruent to
the hypotenuse and a leg of another
right triangle, then the triangles are
congruent. (HL)
Using Corresponding Parts of
Congruent Triangles
• How many triangles are in the next two
figures?
1
3
2
• 15
• 31
Identifying Common Parts
• The idea behind this whole section (as
seen on the last slide) is that triangles
may share sides or angles.
• You need to identify these shared sides
and angles
• Sometimes you have to pull these
triangles apart to see the relationship
Assignment
•
•
•
•
•
Page 261 1-4
Page 262 12,13
Worksheet 4-4
Worksheet 4-6
Worksheet 4-7
Similarity Introduction
• Previously, we learned how to prove
triangles congruent
• Now we will look at how to prove
triangles are similar
• As a reminder, by definition triangles
are similar if they have congruent
angles, and sides which have a uniform
similarity ratio
Angle -Angle Similarity (AA~)
• Postulate: If two angles of one triangle
are congruent to two angles of another
triangle, then the triangles are similar
A
D
B
C
E
Triangle ABC ~ triangle DEF
F
Example
• Are the triangles similar?
B
A
E
45
45
C
D
Yes. Vertical angles are congruent, and angle c and angle
b are congruent (both are 45 degrees). Therefore triangle
AEC is congruent to triangle DEB
Can we write a similarity statement?
No. we don’t know the lengths of any sides to get a ratio
Side Angle Side Similarity
(SAS ~)
• If an angle of one triangle is congruent
to an angle of a second triangle, and the
sides which are connected to each
angle are proportional, then the
triangles are similar
Example
D
A
6
B
9
6
4
C
E
F
Triangle abc is similar to triangle def because angle a is congruent
to angle d, and side ab is proportional to side de, and side ac is
proportional to side df (the same proportion)
We can also conclude that side bc is proportional to side ef
What is the similarity ratio? 2/3
If side EF is 8 long, how long is side BC?
It would be 2/3 of 8 or 5.33
Side-Side-Side Similarity
(SSS ~)
• If the corresponding sides of two triangles are
proportional, then the triangles are similar
• Again, this makes sense. If all sides are in
proportion, then the angles will necessarily be
equal, which is the definition of similar
• No matter how you try to put the second
triangle together, it will only fit one way, and
that way will produce a similar triangle
Find the value of x
12
6
x
8
There are a couple of ways to do this
1) 6/8 = x/12; therefore x = 9
2) 6/x = 8/12; therefore x = 9
Assignment
• Page 455 7-12
• Page 457 24-26
• Worksheet 8-3
Proportions in Triangles
As a review, solve for x
30
X
84/x = 45/15
So x = 28
15
7.5
X/7.5 = 9/6
So x = 11.25
3
6
Side Splitter Theorem
• If a line is parallel to one side of a
triangle and intersects the other two
sides, then it divides the sides
proportionally
Side Splitter Examples
• Solve for X
X
16
Since the lines are parallel, the sides
are proportional. Therefore x/16 =
5/10, or x = 8
5
10
5/2.5 = (x+ 1.5)/x
Therefore x = 1.5
Note: The difference
X+
in this as compared to
what we did before is that we
used to find the total length of the
side. Now we just use a straight
ratio
1.5
X
5
2.5
Corollary to the Side Splitter
Theorem
• If 3 parallel lines intersect 2
transversals, then the segments
intercepted in the transversals are
proportional
• A/B = C/D
C
A
B
D
Example with a Boat :)
•The panels in the sail are
sewn in a parallel pattern
•Solve for X, and solve for Y
•2/x = 1.7/1/7 therefore x = 2
•3/2 = y/1.7 therefore y = 2.55
2
x
3
2
1.7
1.7
y
1.7
Solve for X and Y
•To solve for x:
•X/30 = 15/26
•Therefore x = 17.3
•To solve for y:
•Y/16.5 = 26/15
•Therefore y = 28.6
16.5
15
x
y
26
30
Triangle-Angle-Bisector
Theorem
• If a ray bisects an angle of a triangle,
then it divides the opposite side into two
segments that are proportional to the
other two sides of the triangle
Example
6
5
x
Remember, this angle
is bisected equally, but
it splits the opposite
side proportionallyThe opposite side is
not split exactly in
half
X/6 = 8/5
X = 9.6
8
Or:
X/8 = 6/5
X = 9.6
Example
• Find the value of Y
Y/3.6 = 8/5
Therefore Y = 5.76
3.6
Y
5
8
Or:
Y/8 = 3.6/5
Therefore Y = 5.76
Assignment
• Page 475 9-22
• Page 476 25-35
• Worksheet 8-5
Unit 8 Final
Extra Credit Question
B
A
D
F
Name any congruent
triangles, and the
postulate used to
prove they are
congruent
(2 points each answer)
C
E
G
H