Download Triangle Theorems

DO NOW 12/15:
Triangle ADB is similar to triangle ABC.
Find the value of x.
(HINT: Use the similarity statement!)
5m
10 m
x
TRIANGLE
THEOREMS
Agenda
1.
E x i t T i c ket
C o r r e c t io n s
2.
Tr i a n g l e
Inequality
T h e o r em
3.
Hinge Theorem
4.
Debrief
Learning Target
I can explain
Triangle Inequality
Theorem and
Hinge Theorem,
and relate it to
triangle similarity.
EXIT TICKET
SPAGHETTI TIME! (PART I)
 Break the stick of spaghetti into ANY three parts.
 Measure each of the three parts.
 On your sheet, identify each segment as shortest, middle and
longest based on the measures.
1. Did anyone have an equilateral triangle? Isosceles? Scalene?
2. Add the shortest and middle sides together and compare to the
length of the longest. Record your observations.
 If possible, use the three segments to make a triangle.
Trace the triangle onto a blank/graph piece of paper.
3. Refer back to your answers from #2 above. What was true of the
lengths of the 3 sides that could be made into triangles?
TRIANGLE INEQUALIT Y THEOREM
 In order to form a triangle, the sum of the two smaller sides
of a triangle must be greater than the third side.
Ex. AB + BC > AC
B
A
C
**MUST BE IN NOTES!**
SPAGHETTI TIME! (PART II)
 Once you have your triangle traced on your paper, label the
shortest side AB, the middle side BC, and the longest side AC.
 Use a protractor to measure angle A , angle B and angle C.
Compare the measures.
1. Which measures are biggest? Which measure is smallest? Which
measure is in between?
2. How is this related to the lengths of the sides?
TABLE OF TRIANGLE PARTS
Side
Angle
Smallest
Middle
Longest
What relationship do we notice between the location of
sides and angles with relative measures?
HINGE THEOREM
 The measure of the angles in a triangle are directly related to
the lengths of the segments opposite from them.
Ex. If AC > BC > AB, then angle B > angle A > angle C
A
B
C
**MUST BE IN NOTES!**
DEBRIEF: TRIANGLE THEOREMS
 How are the sides and angles of triangles related?
 How can these two theorems help us when we are trying to
determine if two triangles are similar?
DO NOW 12/16:
Use a similarity statement and
proportions to find the length of y.
SPECIAL
RIGHT
TRIANGLES
Agenda
1.
Embedded
Assessment
Re v i e w
2.
Construct/Bise
ct Equilateral
Tr i a n g l e s ( 3 0 60-90)
3.
Construct
Isosceles
R i g h t Tr i a n g l e s
(45-45-90)
4.
Compare
Ratios
5.
Debrief
Learning Target
I can
determine the
ratios of the
sides of all 4545-90 and 3060-90 right
triangles.
EMBEDDED ASSESSMENT REVIEW
 How far will the judge travel from the start to the 2 nd buoy
 How far will he be at the 2 nd buoy to the finish?
 How far is the 2 nd buoy from the marker?
CONSTRUCT A SCALENE RIGHT TRIANGLE
1.
2.
3.
4.
5.
6.
Draw segment AB of ANY length.
Draw a circle with center point A and radius AB.
Draw another circle with center point B and radius BA .
Label the 2 points of intersection of the two circles C and D.
Draw triangle ABC.
Bisect the triangle by drawing segment CD.
Use the table below to record the side lengths, angle measures
and ratios of the two resulting right triangles.
Shorter Leg
Long/Short
Angle A
Longer Leg
Hyp/Long
Angle M
Hypotenuse
Hyp/Short
Angle C
CONSTRUCT AN ISOSCELES RIGHT
TRIANGLE
1. Draw segment AB of ANY length.
2. Draw the perpendicular bisector of AB
1.
2.
3.
4.
Draw circle A with radius greater than half AB
Draw circle B with radius greater than half AB
Label the two points of intersection of the circles C and D
Connect points CD to bisect AD, marking the point of intersection point M
3. Use the compass to create two congruent segments MA and
MB, and connect AB to form a triangle.
Use the table below to record the side lengths, angle measures
and ratios of the resulting right triangle.
Leg
Leg/Leg
Angle M
Leg
Hyp/Leg
Angle A
Hypotenuse
Hyp/Leg
Angle C
SPECIAL RIGHT TRIANGLES: 30-60-90
 All right triangles with angle measures 30 -60-90 will have
similar side ratios of 1:2:√3
A
2x
60°
x
30°
B
C
x√3
**MUST BE IN NOTES!**
SPECIAL RIGHT TRANGLES: 45-45-90
 All isosceles right triangles with angle measures of 45 -45-90
will have side ratios of 1:1:√2
B
x
x
45°
45°
A
C
x√2
**MUST BE IN NOTES!**
DEBRIEF: SPECIAL RIGHT TRIANGLES
 How are special right triangles related to similarity?
 How is Hinge Theorem and Triangle Inequality related to
special right triangles?
DO NOW 12/17:
Below are two special right triangles.
Use the ratios to determine the missing sides.
TRIG
RATIOS
Agenda
1.
Examples 1-6
2.
R i g h t Tr i a n g l e
Ratios in
Groups
3.
E x a m p l e s 7 - 10
4.
Debrief
Learning Target
I can identify the
opposite, adjacent,
and hypotenuse of
a right triangle, and
use the ratios to
determine triangle
properties.
LESSON 25: TRIGONOMETRIC RATIOS
LABELING A RIGHT TRIANGLE
GROUP 1
GROUP 2
DEBRIEF/EXIT TICKET
 Are ratios easier/harder than similar triangles? The same?
What about it and why?
 What are some applications of these ratios in the real world?
DO NOW 12/18:
Use the ratios of special right
triangles to find the missing
variables.
SIMPLIFYING
RADICALS
Agenda
1.
Fa c to r
Tr e e / P r i m e
Fa c to r i z a t i o n
2.
Adding and
subtracting
radicals
3.
P a c ke t Wo r k
4.
Debrief
Learning Target
I can simplify
and perform
operations with
radicals and use
radicals to solve
problems with
right triangles.
SIMPLIFYING RADICALS
 A radical is another name for a “root” (like “square root”).
 To simplify a radical means to reduce the radicand by
factoring until there are no more perfect “roots” under the
radical sign.
Ex. √75
**MUST BE IN NOTES!**
Ex. √32x 2
OPERATIONS WITH RADICALS
 The key to operating with radicals is to think of them like
variables. You can multiply and divide them together, but you
can only add or subtract if they are like terms.
 Multiplication/Division
√5 × √2 = √10
√12 ÷ √2 = √6
 Addition/Subtraction
4√6 + 3√6 = 7√6
8√3 - √75 = 3√3
** No radicals can be in the denominator!**
4
√3
PRACTICE WITH SPECIAL RIGHT
TRIANGLES AND RADICALS
PROBLEM SET
DEBRIEF/EXIT TICKET
 When are some common situations when we use radicals in
trigonometry?
 When will simplifying radicals be better than using a decimal?
DO NOW 12/19:
Eat a candy cane and take your quiz.
SPECIAL
RIGHT
TRIANGLES
QUIZ
Agenda
1.
Eat Candy
Canes
2.
Ta ke a q u i z
3.
F i n i s h yo u r
p a c ke t / D o
extra credit
wo r k
4.
Debrief
RIGHT TRIANGLES WITH AN ALTITUDE
OR
“GEOMETRIC MEAN”
SPECIAL RIGHT TRIANGLES
DEBRIEF
 Have a safe and relaxing holiday!