# 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 corner-like 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.

Naming and Describing Triangles

• Notation
• Triangle:  $\Delta ABC$
• Angle A:  $\angle A$
• Angle B:  $\angle B$
• Angle C:  $\angle C$

• Describing a triangle
• Sides are ALWAYS opposite angles
• Angles are ALWAYS opposite sides
• Side AB is included between angle A and angle B
• Side AC is included between angle A and angle C
• Side BC is included between angle B and angle C
• Side AB is opposite angle C
• Side AC is opposite angle B
• Side BC is opposite angle A

• Naming triangles
• All polygons, including triangles, can be named starting with any vertex and going around clockwise or counter-clockwise.
• $\Delta ABC$
• $\Delta BCA$
• $\Delta CAB$
• $\Delta ACB$
• $\Delta CBA$
• $\Delta BAC$

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°).
• It’s a cute triangle.
• 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 $\Delta TAX$, $\Delta TXA$, $\Delta AXT$, $\Delta ATX$, $\Delta XAT$, and $\Delta XTA$ 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).