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Unit 2 Notes / Secondary 2 Honors
Day 1: 3.1/3.2 Triangle Angle Theorems and Inequalities
3.1: Pg 212 Triangle Sum Activity: The teacher has a large colored triangle on the board. A student is going to tear
off each corner of the triangle to make a discovery about the sum of the angles in a triangle. Write a sentence
describing the discovery made in the activity:
The Triangle Sum Theorem says:
1.
a.
b.
c.
d.
e.
f.
Write a description for each of the following triangles:
Acute:
Obtuse:
Right:
Scalene:
Isosceles:
Equilateral:
2. Pg 213 Use a straight edge to draw a large scalene triangle in the space below. Label the sides of the triangle
S, M and L for small, medium and large. Use a protractor to measure and record the size of each interior
angle of the triangle and label the angles S, M and L. Compare your results with your partner and the class.
What conclusion can we draw about the relationship between the lengths of the sides of a triangle and the
measure of the interior angles?
3. Pg 217 List the sides from shortest to longest. Complete the problems below, then compare with your
partner.
Pg 219 The remote interior angles of a triangle are the two angles that are non-adjacent to the specified angle.
4. Pg 220. Prove the Exterior Angle Theorem. Work with a partner. Be prepared to explain your reasoning to
the class.
The Exterior Angle Theorem says: The measure of the exterior angle of a triangle is equal to the sum of the
measures of the two remote interior angles of the triangle.
Given: Triangle ABC with exterior angle ∠𝐴𝐶𝐷
Prove: 𝑚∠𝐴 + 𝑚∠𝐵 = 𝑚∠𝐴𝐶𝐷
STATEMENTS
REASONS
2. triangle sum theorem
3. linear pairs are supplementary
5.Subtraction property
5. Pg 221 #14 Solve for x and give the angle measures. Complete problem b if you have time complete d.
6. The Exterior Angle Inequality Theorem says: an exterior angle must be larger than either remote interior
angles. Use the diagram below to discuss this theorem as a class:
3.2. Pg 230 Pasta Activity Sarah thinks any three lengths can represent the lengths of the sides of a triangle.
Sam does not agree. Let’s explore. Take your piece of pasta and break it a two random points so the strand is
divided into three pieces. Measure each of your three pieces in centimeters to the tenths place. Try to form a
triangle from your three pieces of pasta. List your three lengths below and state whether or not the lengths could
form a triangle.
__________________________________________________________________________________________
Random sample of class measurements:
Piece 1 (cm)
Piece 2 (cm)
Piece 3 (cm)
Forms a triangle? (yes/no)
7. With your partner write a hypothesis for what must be true for the 3 lengths to be able to form a triangle.
Be prepared to share your statement with the class.
Is it possible to form a triangle using segments with the following measurements? Sketch a diagram and explain
your answers.
a. 1.9 cm, 5.2 cm, 2.9 cm
b. 152 cm, 73 cm, 79 cm
Pg 233: The Triangle Inequality Theorem states: The sum of the lengths of any two sides of a triangle is greater
than the length of the third side.
Example: If a triangle has two sides measuring 4 cm and 7 cm, what are the possible lengths for the third side?
Day 2: 3.3/3.4 Special Triangles
1. What is the Pythagorean Theorem? (Write out the formula using the words “leg” and “hypotenuse”)
Does it work for all triangles? Explain.
2. a. Use a ruler to measure the two legs of the triangle to the right.
What kind of triangle is the triangle to the right?
b. Use a protractor to measure the two acute angles in the triangle.
What is another name for the triangle?
c. Use your ruler to measure the length of the hypotenuse. Do you get an integer answer?
Write down the decimal answer, to the nearest tenth of a centimeter, for the length of the hypotenuse.
Compare your estimate for the length of the hypotenuse with your neighbor. Are your answers exactly
the same?
Would we expect all the member of the class to have exactly the same estimate?
How could we get an exact length for the hypotenuse? (Hint: look at step #1 above)
Use this method to find an exact/reduced root length for the length of the hypotenuse. Record this length
on your diagram above.
3. Use Pythagorean Theorem to find the length of the hypotenuse for a 45°-45°-90° triangle with 6 inch legs.
What pattern do you notice in the answers to #2 and #3?
Sketch a diagram of a 45°-45°-90° triangle with side lengths 𝒙 long. Record the length of the hypotenuse
on the diagram.
4.
Use Pythagorean Theorem to find the missing side lengths of the triangles. Answer in reduced root form.
x
x
8
5
15
5
b = 2√3 , c = 6
5.
Are any of the triangles above a 45°-45°-90° triangle? Explain.
6.
Find all the missing side lengths for the triangles below.
x
6 2
45
x
x
6
6
7. An equilateral triangle is shown below. One angle has been bisected with a perpendicular bisector. Record the
angle measures of the angles in the 2 triangles formed.
a. What are the 3 angle measurements of the newly formed triangles?
̅̅̅̅ ?
b. What is the length of 𝐴𝐷
̅̅̅̅ ?
What is the length of 𝐵𝐷
c. Use Pythagorean Theorem to find the length of ̅̅̅̅
𝐶𝐷 .
What pattern do you notice in the side lengths of the 30°-60°-90° triangles?
Compare your answer with your partner.
Sketch a diagram of a 30°-60°-90° triangle with a Short Leg 𝒙 long. Record the length of the Long Leg and the
Hypotenuse on the diagram.
8.
Use the above investigation to find the missing side lengths of following triangle.
First label the side lengths: Short Leg, Long Leg and Hypotenuse then find the missing lengths.
x
3
30 
y
9.
What is a rule for finding the side lengths of a 30°-60°-90° triangle in relationship to the short leg:
Hypotenuse = _____________________
Long Leg = ________________________
10. Find all missing side lengths in the following diagrams:
8
y
60 
y
30
60
x
x
a)
6
b)
11. Why are the 45°-45°-90° triangle and the 30°-60°-90° triangle “special”?
Day 3: 4.1/4.2 Similar Triangles
1) Dilation vocabulary to know:
Scale Factor
Point of dilation
Corresponding parts
Proportional
Larger or Smaller – Carnegie p266 Problem 3
Consider ∆𝑮𝑯𝑱 shown on the coordinate plane. You will dilate the triangle by using the origin as the center
and by using a scale factor of 2.
2) How will the distance from the center of dilation to a point
on the image of ∆𝐺′𝐻′𝐽′ compare to the distance from the
center of dilation to a corresponding point on ∆𝐺𝐻𝐽? Explain
your reasoning.
3) How do the coordinates of the image compare to the
coordinates of the pre-image?
1
4) ∆𝑀𝑂𝐵, with vertices 𝑀(0, −3), 𝑂(−12,6) and 𝐵(5,4), has been dilated by a factor of , using the origin as
3
the center of dilation. What are the new vertices?
Making a Shadow – Carnegie p262 Problem 2
Consider ∆𝑨𝑩𝑪, ∆𝑫𝑬𝑭, and point 𝒀. Imagine that point 𝒀 is the flashlight and ∆𝑫𝑬𝑭 is the shadow of ∆𝑨𝑩𝑪.
This creates Similar Triangles.
5) Use your measurements to express the
following ratios as fractions then as
decimals.
𝐷𝐸
=
7.1
𝐴𝐵
4.8
𝐸𝐹
=
𝐵𝐶
9
𝐷𝐹
=
𝐴𝐶
6)
Use a protractor to measure the
corresponding angles in the triangles.
What can you conclude?
2.8
1.9
3.6
PROVING TRIANGLES ARE SIMILAR
What do we always know about similar triangles?
*The corresponding angles are ______________________. The corresponding sides are ________________.
So, if ∆𝐶𝑂𝑊~∆𝑃𝐼𝐺, what do we know?
AA Similarity Theorem
SAS Similarity Theorem
SSS Similarity Theorem
Determine whether the given triangles are similar. If so, use symbols to write a similarity statement AND state
which theorem you used.
Carnegie p279 #8
Carnegie p278 #6
Two triangles with side lengths that can be written:
𝐴𝐵 𝐵𝐶 𝐴𝐶
=
=
𝑋𝑌 𝑌𝑍 𝑋𝑍
Carnegie p276 #6
Carnegie p276 #7
GIVEN: ⃡𝐴𝐵 ∥ ⃡𝐸𝐷
PROVE: ∆𝐴𝐵𝐶~∆𝐷𝐸𝐶
Day 4: 4.3/4.4 Triangle Proportionality Theorems
4.3: Pg 328
Applying the Angle Bisector/Proportional Side Theorem
1. The Angle Bisector/Proportional Side Theorem says: A bisector of an angle in a triangle divides the opposite
side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle.
Path E bisects the angle formed by path A and path B.
Path A is 143 feet long.
Path C is 65 feet long.
Path D is 55 feet long.
̅̅̅̅ bisects ∠𝐶. Solve for DB.
𝐶𝐷
Find the length of path B.
Pg 328 Applying the Triangle Proportionality Theorem
2. The Triangle Proportionality Theorem says: If a line parallel to one side of a triangle intersects the other two
sides, then it divides the two sides proportionally.
Are the triangles similar?
Justify
Now solve for FG using similar triangles.
GE ∥ HD, DE = 30, EF = 45, GH = 25, FG = ?
Solve using the theorem above.
Pg 329. Write the CONVERSE of the Triangle Proportionality Theorem.
3. The converse of the triangle proportionality theorem can be used to test whether two lines segments are
parallel.
Given: DE = 33, EF = 11, GH = 66, FG = 22.
̅̅̅̅ ∥ ̅̅̅̅
Is 𝐷𝐻
𝐸𝐺 ? Justify using proportions.
Pg 329 Applying the Proportional Segments Theorem.
4. The Proportional Segments Theorem says: If three parallel lines intersect two transversals, then they divide the
transversals proportionally.
a) Given: 𝐿1 ∥ 𝐿2 ∥ 𝐿3 , AB = 52, BC = 26, DE = 40, find EF.
b) Given: 𝐿1 ∥ 𝐿2 ∥ 𝐿3 , AB = 90, EF=15, DE = 75, find BC.
Pg 330 Applying the Triangle Midsegment Theorem.
5. The Triangle Midsegment Theorem says: The midsegment of a triangle is parallel to the third side of the
triangle and half the measure of the third side of the triangle.
a. Given E is the midpoint of FD, G is the midpoint of FH, and DH = 15,
find the measure of EG.
b. What do you know about ∠𝐸 and ∠𝐷? Why?
Pg 330. 6. Using the Right Triangle/Altitude Theorem. If an altitude is drawn from the vertex of a right angle to
the hypotenuse, then three similar right triangles are formed! Use proportions to solve for the missing lengths.
Can you see the three similar triangles?
Hint: Label the sides of the triangles S, M and L, then set up the proportion. Solve for all variables.
Day 5: 4.5/4.6 Similar Triangle Applications
What is the converse of the Pythagorean Theorem?
1) If a triangle has sides that measure, 12, 16, and 20, would it be a right triangle? Justify your answer
with calculations.
How Tall Is That Oak Tree? Carnegie p 320 Problem 2 #1, #3 and How Wide Is That Creek? Carnegie
p322 Problem 3 #1
1) You go to the park and place a mirror on the ground so you can see
the top of a tree. You then gather enough information to calculate
the height of one of the oak trees. The figure shows your
measurements.
a. Are the triangles similar? How do you know?
b. Calculate the height of the tree.
2) Stacey notices that another tree casts a shadow and suggests that
you could also use shadows to calculate the height of the tree. She
lines herself up with the tree’s shadow so that the tip of her shadow
and the tip of the tree’s shadow meet. She then asks you to
measure the distance from the tip of her shadows to her, and then
measure the distance from her to the tree. Finally, you draw a
diagram of the situation as shown to the right. Calculate the height
of the tree. Explain your reasoning.
a. Are the triangles similar? How do you know?
b. Calculate the height of the tree.
3) You stand on one side of the creek and your friend stands directly across the creek from you on the
other side as shown in the figure. Your friend is standing 5 feet from the creek and you are standing
5 feet from the creek. You and your friend walk away from each other in opposite parallel
directions. Your friend walks 50 feet and you walk 12 feet.
a. How do you know the triangles formed are similar?
b. Calculate the distance from your friend’s starting point to your side of the creek.
c. What is the width of the creek? Explain your reasoning.