Download Classifying Triangles by Angle Measure

Triangles Packet
DATE
HOMEWORK
DATE
HOMEWORK
1
1.
Basic Triangle/Angle Info
Define the following terms and draw a picture of each:
Angle
Acute Angle
Obtuse Angle
Right Angle
Supplementary Angles
Complementary Angles
Congruent Angles
Vertical Angles
Adjacent Angles
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Basic Triangle/Angle Info
Define the following terms and draw a picture of each:
Definition and diagram
Extra information
Right Triangle
Isosceles Triangle
Scalene Triangle
Equilateral Triangle
Equiangular Triangle
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Classifying Triangles
Classifying Triangles by Angle Measure:
A ____________________triangle has one angle equal to 90 degrees.
EXAMPLE: (draw a picture!)
A____________________ triangle has all three angles measuring less than 90 degrees.
EXAMPLE: (draw a picture!)
A____________________ triangle has all exactly one angle measuring more than 90 degrees.
EXAMPLE: (draw a picture!)
Classifying Triangles by Side Length:
A____________________ triangle has two equal sides (and the two angles opposite those
sides) that are equal.
EXAMPLE: (draw a picture!)
A____________________ triangle has no two sides (and no angles) congruent.
EXAMPLE: (draw a picture!)
A____________________ triangle has all three sides (and angles) equal in measure.
EXAMPLE: (draw a picture!)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Two triangles are called ____________________ if they have the same angles (same shape).
Two triangles are called ____________________if they have the same angles and the same
side lengths (same shape and size).
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Basic Triangle/Angle Info
The sum of the interior angles of a triangle is always _____________________.
Examples: Find x for each.
x
x
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Angles that form a straight line are called a _____________________________
and together measure ____________. These angles are also called
_________________________________________.
Examples: Find the value of x.
x
Calculate the measure of angles BCD and ABC. Explain your answers!
𝑚 < 𝐵𝐶𝐷 =__________ because…
𝑚 < 𝐴𝐵𝐶 =__________ because…
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Vertical Angles
____________________________ are the angles opposite each other when two
lines cross and are always ___________________________.c
Examples: Find the value of the variables used in each diagram.
………………………………………………………………………………………………………………………………………………………….’
Examples: Find the measure of the requested angle based on the information given
for each problem.
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Practice
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Practice
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Practice
3.
4.
5. Find the missing value.
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Practice
(d)
(e)
(f)
CHALLENGE: Find the value of x in each diagram.
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Triangle Inequality Theorem
Spaghetti Triangles
Follow these directions:
1. Carefully measure and cut (break) lengths of spaghetti noodles according to the following amounts:
1 inch; 2 inches; 3 inches; 4 inches; 5 inches; 6 inches; 7 inches; 8 inches
2. Try to make a triangle with the given lengths in the chart. Write yes or no in the appropriate column.
3. Generalize your findings by writing a “rule” on how to tell if three given lengths can form a triangle.
4. Answer the questions at the bottom of the page.
Triangle? Yes or No
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1
3
3
2
1
2
5
2
5
3
4
5
7
4
3
6
5
6
5
5
6
8
5
4
7
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My rule for determining if three given lengths can form the sides of a triangle:
Use your rule to determine if the following lengths could form a triangle. Write yes or no.
(a) 34, 40, 83 ________________
(b) 5, 33, 51 _______________
(c) 19, 31, 50 ________________
(d) 13, 18, 29 ______________
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Triangle Inequality Theorem
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Triangle Inequality Practice
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Isosceles Triangles
An ____________________triangle has two equal sides and the two
angles opposite those sides are equal. An isosceles triangle can be
_______________, _______________, or ________________.
EXAMPLE: Find the value of the variables in each problem.
x°
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Equilateral Triangles
An ____________________triangle has all sides equal and all angles equal.
EXAMPLE: Find the unknown values.
EXAMPLES: Find the value of the x and/or y in each diagram.
x°
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Practice
Find the value of x.
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Refer to the figures at right. Choose the
statement that is true about the given values.
A. The value in column A is greater.
B. The value in column B is greater.
C. The two values are equal.
D. The relationship cannot be determined
from the given information.
1.
2.
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Right Triangles
A ______________________________ is a triangle with one right angle. A right triangle can be
isosceles OR scalene, but not equilateral.
The ______________________________ is the name of the
longest side of a right triangle that is located directly opposite of the right angle.
The two shorter sides of the triangle are called the _____________ of the triangle.
The ______________________________________ relates the lengths of the sides of a right triangle. The
formula is:
EXAMPLE: Find the length of the third side of the triangle.
EXAMPLE: Do the following lengths form a right triangle? Explain.
EXAMPLE: A fishing rod is used to catch ducks in a fairground game.
The rod is 1 m long. A string with a ring is tied to the end of the rod. The
length of the string is 0.4 m. When the ring is level with the lower end of
the rod (as shown in the diagram), how far is the ring from that end of the
fishing rod?
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Practice
1. Find the length of the unknown side of the triangle.
2. Is the triangle below a right triangle? Why or why not?
3. The whole numbers 3,4,5 are called a Pythagorean triple because 32+42=52. A triangle with sides of length 3
cm, 4 cm, and 5 cm is a right triangle. Use Pythagoras’ Theorem to determine which sets of numbers below are
also Pythagorean triples.
a) 15, 20, 25
b) 10, 24, 16
c) 11, 22, 30
d) 6, 8, 9
4. Calculate the lengths of the diagonals of the rectangle shown.
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Practice
5. The diagram shows a wooden frame that is to be part of the
roof of a house.
a) Use Pythagoras’ Theorem in triangle PQR to find the
length PQ.
b) Calculate the length QS.
c) Calculate the total length of wood needed to make the frame.
6. Which of the triangles below has the longer diagonal? Explain your answer.
7. Find the area of each triangle shown (you will have to find the height FIRST).
8. The width of a rectangle is 5 cm and the length of its diagonal is 13 cm.
a) What is the measure of the LENGTH of the rectangle?
b) What is the area of the rectangle?
9. A hiker walks 200km due north, then 300km due west. How far is the hiker directly from where they started?
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Congruent Triangles
There are certain methods by which we KNOW two triangles are congruent.
** If each side of a triangle is congruent to a corresponding side of another triangle, then the triangles MUST be
congruent. In the example above: ∆_____ ≅ ∆_____ by _______
** If two sides and the included angle of one triangle are congruent to two sides and the included angle of
another triangle, then these two triangles are congruent. In the example above: ∆_____ ≅ ∆_____ by _______
** If two angles and the non-included side of a triangle is congruent two angles and the non-included side of
another triangle, then the triangles MUST be congruent. In the example above: ∆_____ ≅ ∆_____ by _______
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** If two angles and the included side of one triangle are congruent to two angles and the included side of
another triangle, then these two triangles are congruent. In the example above: ∆_____ ≅ ∆_____ by _______
Directions: Check which congruence postulate you would use to prove that the two triangles are congruent.
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Congruent Triangles
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Congruent Triangles
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Congruent Triangles
Use the given information to mark the diagram appropriately. Name the triangle congruence (pay attention to
proper correspondence when naming the triangles) and then identify the Theorem or Postulate (SSS, SAS, ASA,
AAS, HL) that would be used to prove the triangles congruent. If the triangles cannot be proven congruent,
state “not possible.”
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Similarity
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Similar Triangles
Two triangles are ___________________ if the angles of one triangle are equal to
the corresponding angles of the other. In other words, similar triangles have the
same shape, but not the same size. In _________________________ triangles,
ratios of corresponding sides are equal (in other words, the side lengths are
proportional).
EXAMPLE:
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Similar Triangles
Use the following figure to answer the questions below. Triangle is not drawn to scale.
Given: ABC is similar to DEF
Question #1) In the above diagram, the triangles are similar. EF = 6 and BC = 2. What is the length of DE if
AB is 3?
Question #2) In the above diagram, the triangles are similar. AB = 8 and AC = 7. What is the length of DF if
DE is 16?
Question #3) In the above diagram, the triangles are similar. DF = 8 and BC = 3. What is the length of EF if
AC is 2?
Question #4) In the above diagram, the triangles are similar. AB = 1 and AC = ½. What is the length of DF if
DE is 10?
Question #5) In the above diagram, the triangles are similar. The perimeter of ABC = 144 and BC = 24.
What is the perimeter of DEF if EF = 96?
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Similar Triangles
How do you know if two triangles are similar?
State if the triangles in each pair are similar. If so, state how you know they are similar and complete the
similarity statement.
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Practice
2. Triangles AFE and ACB are similar.
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Right Triangle Trigonometry
__________________________________ is
the study of three sided figures and the relationships between their sides and angles.
EXAMPLE: Find all 6 trigonometric functions for the right triangle below.
sin 𝜃 =
csc 𝜃 =
cos 𝜃 =
sec 𝜃 =
tan 𝜃 =
cot 𝜃 =
EXAMPLE: Find the value of all variables in each of the following:
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Right Triangle Trigonometry
Find the measure of each angle in
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Right Triangle Trigonometry
Solve for each variable.
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Right Triangle Trigonometry
Find the measure of each angle indicated.
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Right Triangle Trigonometry
Find the measure of each side indicated.
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Right Triangle Trigonometry
Solve each triangle (find all the missing side and angle measures). Round your
answers to the nearest tenth.
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Right Triangle Trigonometry
Find the value of x:
Find the value of x:
Find the value of x and y:
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Right Triangle Trigonometry
Using the following figure, fill-in the chart. Show your work on the back of this paper. Calculators are
allowed, but write out the steps. Round your answers to two significant digits past the decimal point.

c
b

a
1
a
5
2
b
30
3
13
5
6
10
45
25
25
80
15
33
8

17
6
9

75
4
7
c
10
33
125
37
0.75
23.9
7/19
55.7
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Right Triangle Trigonometry
Draw a diagram and use trigonometry to solve the following.
1. A ten-foot ladder is leaned against the side of a house in such a way that it makes an angle of 65o with the
ground.
a) How high up the house does the ladder reach? (Round your answer to the nearest tenth.)
b) Is the ladder long enough to reach the second story window which is 15 feet above the ground? If
not, how long a ladder WOULD be needed in order to reach the second story window (in the event of a
fire)?
2. A camper is hiking and is standing on top of a 400 foot cliff enjoying the view. He looks down and views a
bear at a 37o angle of depression. How far is the bear from the base of the cliff? (Round your answer to the
nearest foot, and disregard the height of the camper in your calculations.)
3. A girl is flying a kite and lets out 250 feet of string. If she sights the kite at a 43o angle of elevation, what is
the height of the kite? (Round your answer to the nearest tenth, and disregard the height of the girl in your
calculations.)
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Right Triangle Trigonometry
1. A damsel is in distress and is being held captive in a tower. Her knight in shining
armor is on the ground below with a ladder. When the knight stands 15 feet from
the base of the tower and looks up at his precious damsel, the angle of elevation to
her window is 60 degrees. How long does the ladder have to be?
2. You are 200 yards from a river. Rather than walking directly to the river, you walk
400 yards along a straight path to the river’s edge. Find the acute angle between
path and the river’s edge.
3. A 12 meter flagpole casts a 9 meter shadow. Find the angle of elevation of the sun.
4. Suppose you’re flying a kite, and it gets caught at the top of the tree. You’ve let out
all 100 feet of string for the kite, and the angle that the string makes with the ground
is 75 degrees. Instead of worrying about how to get your kite back, you wonder.
“How tall is that tree?”
5. Suppose that Mike and Dave are making measurements for the road-paving crew.
They need to know how much the land slopes downward along a particular stretch
of road. Dave walks 80 feet from Mike and holds up a long pole, perpendicular to
the ground, that has markings every inch along it. Mike looks at the pole through a
sighting instrument. Looking straight across, parallel to the horizon, Mike sights a
point on the pole 50 inches above the ground- call it point A. Then Mike looks
through the instrument at the bottom of the pole, creating an angle of depression.
Which is the angle of depression or slope of the road, to where Mike is standing?
6. A submersible traveling at a depth of 250 feet dives at an angle of 15º with respect
to a line parallel to the water’s surface. It travels a horizontal distance of 1500 feet
during the dive. What is the depth of the submersible after the dive?
7. A fire department’s longest ladder is 110 feet long, and the safety regulation states
that they can use it for rescues up to 100 feet off the ground. What is the maximum
safe angle of elevation for the rescue ladder?
8. Brothers Bob and Tom Katz buy a tent that has a center pole 6.25 feet high. If the
sides of the tent are supposed to make a 50° angle with the ground, how wide is the
tent?
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