Triangles
B
Triangles: 3 sided polygons
Name by angles
This is ABC
The sum of the three angles
of a triangle is 180°.
A + B + C = 180°
a
c
A
sides: AB, BC, AC
or a, b, c (based on opposite angle)
C
b
Vertices: A, B, C
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Name the vertices:
Name the sides:
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Label sides with a single letter based on angles
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Review:
acute angle – measures greater than 0° and less than 90°
Right angle – measures exactly 90°
Obtuse angle – measures greater than 90° and less than 180°
Straight angle – measures exactly 180°
Classify triangles by angle types:
1. Acute triangle –
must have all three angles acute
2. Right triangle –
must have 1 right angle
3. Obtuse triangle –
must have 1 obtuse angle
Questions:
1. Give one example of the measures of three angles for an acute triangle. ______, ______, _______
2. Can you have more than 1 right angle in a triangle? Why or why not?
3. Can you have more than 1 obtuse angle in a triangle? Why or why not?
4. Can you have all three angles of a triangle equal? If so, what would the measure of each angle be?
Classify triangles by number of equal sides
1. Scalene triangle – no equal sides (no equal angles either)
2. Isosceles triangle – at least 2 equal sides
(would also have 2 equal angles)
Equal sides are called legs.
The non-equal side is called the base.
The angle between the legs is called the vertex angle.
3. Equilateral triangle – all 3 sides equal
(would also have all 3 angles equal)
Special situations:
Right triangles have specific names for the sides.
The longest side is always opposite the right angle and is called the hypotenuse.
The shorter sides are called legs.
hypotenuse
leg
leg
Practice:
1. Name the type of triangle shown by the angle measure. If it is impossible to have a triangle with the given
measures, write no triangle. Use acute, right, obtuse, equiangular as your choices for answers.
a. 30, 20, 130 _______________
b. 45, 45, 90 _________________
c. 60, 60, 60 _________________
d. 85, 45, 50 __________________
e. 50, 30, 80 ________________
f. 74, 24, 82 __________________
2. Name the three types of triangles based on side measure.
a. __________________, ___________________, ___________________
In a triangle, the two shorter sides must always add together to give a sum which is more than the largest side.
Example: 4, 7 , 6 could be a triangle because 4 + 6 > 7
3, 9, 14 could not be a triangle because 3 + 9 < 14
You can determine if a triangle is right, acute or obtuse by putting the sides into the Pythagorean Theorem.
If the sides fit to make a true statement, it is a right triangle.
If the hypotenuse squared is “too big”, or is larger than the sum of the squares of the shorter sides, it is
obtuse. (The legs much stretch out to reach the ends of the hypotenuse, making > 90°)
If the hypotenuse squared is “too small”, or is smaller than the sum of the squares of the shorter sides, it
is obtuse. (The legs must lean in closer to reach the ends of the hypotenuse, making < 90°)
b. Tell which of the following could be sides of triangles. Then, for those which are triangle sides, tell whether
the triangle will be right, acute, or obtuse.
1. 4, 8, 14
4.
5, 7, 8
2. 5, 12, 13
5.
6, 5,
3. 7, 3, 6
6.
8, 9,
61
21
3. How could you describe the triangle to the right? Give classification by angle and sides.
_____________________, _____________________
B
4. Use the figure at the right to answer the following questions:
a. Identify an equilateral triangle. ______________
b. Name a right triangle.____________.
D
S
c. Name the hypotenuse of the right triangle. __________
d. Name an isosceles triangle. _____________
e. Which sides of the isosceles triangle are equal? ______________ and ______________
f. Name three angles which must be equal. ______, _____, _____
What is the measure of each of these angles? __________
5. Complete with sometimes, always, or never.
a.
b.
c.
d.
e.
f.
g.
h.
Equilateral triangles are _______________ isosceles.
Scalene triangles are _________________ isosceles.
Right triangles are ___________________ acute.
Acute triangles are ___________________equilateral.
Obtuse triangles are ___________________ scalene.
Equiangular triangles __________________ acute.
Obtuse triangles are ___________________isosceles.
Isosceles triangles are __________________ equilateral.
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E
6. Use the figure at the right to answer the following questions.
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a. Identify an acute triangle. _________________
C
b. Name the hypotenuse. ________________
B
c. Name the isosceles triangle. __________
d. Name the vertex angle of the isosceles triangle. ________
e. Name the side opposite C. ______________
f. Name the angle opposite side MB. ___________
g. Name the base angles of the isosceles triangle. _______________
h. Name the base of the isosceles triangle. ________________
i. Name the legs of the right triangle. ______________________
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