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2.1 – What is a Triangle?
Objectives
 Define a triangle.
 Label the parts of a triangle: sides, angles, and vertices.
 Use proper notation to name triangles.
 Group triangles by their angle measurements (acute, right, and obtuse).
 Group triangles by their side lengths (equilateral, isosceles, and scalene).
 Use the triangle inequality theorem to build triangles.
Key Terms
 Acute Triangle – A triangle in which all three interior angles are acute.
 Angles – The cornerlike spaces where the sides of a triangle meet.
 Equilateral Triangle – A triangle with three sides of equal length.
 The three angles of an equilateral triangle each measure 60°.
 All equilateral triangles are isosceles.
 All equilateral triangles are acute.
 The word equilateral comes from the Latin prefix aequi, meaning “equal” and the Latin root lateralis, which means “belonging to the side.”
 Isosceles Triangle – A triangle that has at least two congruent sides.
 The angles opposite these sides are also congruent.
 Some isosceles triangles are equilateral, not all.
 Isosceles triangles can be acute, obtuse, or right.
 The word isosceles comes from the Greek isos, meaning “equal” and the Greek skelos, which means “leg.”
 Obtuse Triangle – A triangle that has one angle measuring more than 90°.
 Right Triangle – A triangle that contains a right angle.
 Scalene Triangle – A triangle in which all three sides have different lengths.
 Scalene triangles can be acute, obtuse, or right.
 Sides – The three line segments that form a triangle.
 Triangle – A polygon with three sides.
 A triangle is often shown with the symbol, followed by the three letters that label the vertices.
 A figure formed by 3 line segments that connects 3 noncollinear points.
 The prefix tri means three. So the word triangle means three angles.
 Vertices – The points at which the sides of a triangle meet.
Review
Angle Measures 
 Important angle measures to remember

Notes
Triangles 
 Characteristics of a Triangle
 Closed, plane (2 dimensional) figures
 Closed means no gaps or holes in the figure
 Have 3 sides
 All sides are line segments (straight and measurable)
 Segments connect 3 noncolliear points

 Parts of a Triangle
 Pay very close attention to the symbols and notation.

 Triangles Sorted by Side Lengths
 Equilateral triangles have all congruent sides.
 ALL equilateral triangles are isosceles triangles (because at least 2 sides are congruent).
 All equilateral triangles are acute (angles are 60° each).
 Isosceles triangles have at least two congruent sides.
 Can be obtuse, right, or acute triangles.
 Some isosceles triangles are also equilateral (but only if they have all 3 congruent sides).
 Scalene triangles have no congruent sides.
 Can be obtuse, right, or acute triangles.

Classification and Confirmation of Triangles 
 Classifying Triangles
 The sum of interior angles of a triangle measure 180 degrees
 Acute triangles have all little angles (the largest angle is less than 90°).
 Right triangles have one angle that is just right (the largest angle is exactly 90°).
 It’s a perfect triangle.
 You can only have ONE right angle in a triangle (as the other two angles will be less than 90°, adding up to a total of 180°).
 Obtuse triangles have one very large angle (the largest angle is greater than 90°).
 It’s an obese triangle.
 You can only have ONE obtuse angle in a triangle (as the other two angles will be less than 90°, adding up to a total of 180°).

 Is it a Triangle?
 Do the two shortest sides add up to more than the 3rd (longest) side? If so, it’s a triangle!
 ex. A triangle with sides 2, 5, and 1
 Step 1: Put the sides in order from smallest to greatest: 1, 2, and 5
 Step 2: Add up the two shorter sides (1 + 2 = 3).
 Step 3: Do those two shorter sides (3) add up to more than the largest side (5)? Is 3 > 5 ?
 No! So, this is NOT a triangle!
 ex. A triangle with sides 2, 5, and 4
 Step 1: Put the sides in order from smallest to greatest: 2, 4, and 5
 Step 2: Add up the two shorter sides (2 + 4 = 5).
 Step 3: Do those two shorter sides (6) add up to more than the largest side (5)? Is 6 > 5 ?
 Yes! So, this IS a triangle!
 Triangle Inequality Theorem
 The sum of the lengths of any two sides of a triangle is greater than the length of the 3rd side.
 To find the possible range of the 3rd side of a triangle, when given the other 2 sides, set up the following problem:
 Large # (minus) Small # < x < Large # (plus) Small #
 x represents all the possible side lengths between the small and large numbers.
 Ex. Given sides 4 and 7, what is the possible range of the length of the 3rd side of this triangle?
 7 is the large number
 4 is the small number
 7 – 4 < x < 7 + 4
 3 < x < 11
 So, the 3rd side of the triangle is a number greater than 3 (such as 3.1) but a number less than 11 (such as 10.99).
 The integers would be 4, 5, 6, 7, 8, 9, & 10.

Examples
 Ex 1. What are the valid names for the given triangle?
Answer: Since names are denoted by vertices, M and Y are NOT part of the triangle’s name.
The names are , , , , , and

 Ex 2. Classify the following triangle as acute, obtuse, or right.
Answer: Since the largest angle is 70°, which is acute, then the triangle is classified as acute.

 Ex 3. Classify the following triangle.
Answer: Two sides have identical hash marks, so this is an isosceles triangle. It also has an angle of 120°, so it is an obtuse triangle.

 Ex 4. Classify the following triangle.
Answer: The largest angle is 102, so this is an obtuse triangle. All three sides have different lengths, so this is also a scalene triangle.

 Ex 5. The segments shown below could form a triangle.
Answer: Put the sides in order from smallest to largest: 7, 9, 16.
Add the smaller two sides: 7 + 9 = 16.
Ask yourself, “Do these two sides add up to a number LARGER than the largest side?” If yes, it’s a triangle. If no, it’s not a triangle.
7 + 9 > 16 is false. Because 16 is not greater than 16. So, this is NOT a triangle.

 Ex 6. A triangle has two sides of lengths 7 and 9. What is the range of the length of the third side?
 Answer: Remember the rule: Large – Small < x < Large + Small
 9 – 7 < x < 9 + 7
 2 < x < 16
 The length of the 3rd side, x, could be 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, or 15 (or any number between 2 and 16).

Important!
Practice (Apex Study 2.1)
 Practice: Pgs 8, 12, 20, 21, 23, 24, 26
 Sorting tool: Pgs 2, 9, 11, 13, 15, 25
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