3.1 Definition of a Triangle and its Classifications...Chapter 3: Introduction to Triangles 3.1 Definition of a Triangle and its Classifications Definition: A _____ is a 3 sided polygon.

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Chapter 3: Introduction to Triangles

3.1 Definition of a Triangle and its Classifications

Definition: A _______________ is a 3 sided polygon. - A _____________ is a closed figure which is the union of line segments. - A triangle also has 3 interior angles which always add to ____________.

Labeling a triangle: Capital letters are used for the vertices. The same letters in lower case are used to represent the sides opposite those vertices.

**Angles of the triangle are written using the single vertex letter or with three letters. **Sides can also be written by their line segment name.

A

b c

B C a

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A triangle can be classified by its angles:

1. An _____________ TRIANGLE has 3 acute angles.

2. An ______________ TRIANGLE has one obtuse angle. (Why only one?)

3. A _______________ TRIANGLE has one right angle. (Again, why only one?)

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We can also classify a triangle based on the number of congruent sides it has.

Classifying a triangle by its sides: 1. A _________________ TRIANGLE has no congruent sides & therefore no congruent angles. Draw examples. Mark the sides and angles accordingly.

Please note the following interesting fact about the triangle:

• A Triangle always has as many congruent angles as it has congruent sides.

So knowing the angles can help you classify the triangle by its sides too.

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2. An ___________________ TRIANGLE has two congruent sides and two congruent angles. Draw examples. Mark the sides and angles accordingly.

3. An ______________________ TRIANGLE has three congruent sides and three congruent angles. Draw an example. Mark the sides and angles accordingly.

What is the measure of each angle of the equilateral triangle? Why?

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This triangle can also be called an __________________ triangle, since the 3 angles are congruent. You try: EX 1: Classify a triangle with angles of 40°, 60° and 80°. EX2: Classify a triangle with angles of 120°, 30°and 30° EX 3: Classify a triangle with angles of 25°, 90° and 65°. EX 4: Classify a triangle with angles of 100°, 60° and 20°. EX 5: Classify a triangle with angles of 51°13’ and 70°25’. EX 6: Classify a triangle with angles of 90° and 45°. EX 7: Classify a triangle with angles of 60° and 59°60’.

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Some triangles have names for particular parts: The Right Triangle: - has a right angle

- symbolized with the small box in the right angle

- The side across from the right angle is the ________________. **It is always the longest side of the right triangle.

- The remaining two sides are the _______________. **They are always perpendicular to each other forming the right angle.

Label the parts of the right triangle below if AC ⊥ CB.

A

C B

Since ∠C is the right angle, the other two angles of a right triangle must be acute. Why?

What angle pair name can you give these two angles?

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The Isosceles Triangle:

- has 2 congruent sides called the ______________

- the non-congruent side is the ______________

- the two angles that share the base are called the _______________ ______________

**These angles are congruent.

- the angle formed by the legs is the _____________ angle.

Label the parts of the Isosceles Triangle below, if AB ≅ AC.

A

B C

You try: 1. If the vertex angle is 106°, 2. If a base angle is 68°, what is each base angle? what is the vertex angle?

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3.2 Interior Angles of a Triangle

• The sum of the 3 interior angles of any triangle is _________ degrees

• A triangle can have only 1 right angle

• A triangle can have only 1 obtuse angle

• If the triangle is a right triangle, then the remaining two

angles must add up to the remaining 90 degrees. In other words, the acute angles of a right triangle are ___________________.

• If the triangle is equilateral, then it’s also equiangular.

(Remember that a triangle always has as many congruent angles as sides.) As a result, each angle of an equilateral triangle measures ________ degrees.

• If the triangle is an isosceles right triangle then each acute

base angle measures _________.

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• The sum of the angles of a quadrilateral is ________ degrees.

- take any quadrilateral and draw a diagonal. The quadrilateral is now a pair of triangles, each having 180 degrees.

You try: 1. Two angles of a triangle are 78° 2. Two angles of a triangle are 24°13’ and 45°. What is the measure of and 36°24’. What is the measure of the third angle? the third angle? Then classify the triangle. Then classify the triangle. 3. The angles of a triangle are represented by (3x + 1)°, (4x - 12)° and (7x + 9)°. Solve for x, find the measure of each angle and then classify the triangle.

180°

180°

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3.3 Exterior Angles of a Triangle

When one side of a triangle is extended, the angle between that extension and the adjacent side is known as an ______________ Angle. (means “outside.”)

In the diagram above, ∠1 is an exterior angle. The measure of an exterior angle of a triangle equals the sum of the two angles inside the triangle that are NOT adjacent to it… the two interior angles that don’t share a side with the exterior angle. In the diagram above, that makes m∠1 = m∠2 + m∠3. What is the measure of exterior ∠1 if ∠2 = 46° and ∠3 = 77°?

1 2

3

1 2

3

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Look at ΔABC below.

How many exterior angles does a triangle have?

What are the degree measures of the exterior angles at A and C in the above diagram? Draw and label the angles.

Below, draw examples of an obtuse triangle and a right triangle with their interior and exterior angles.

What can be concluded about the sum of the exterior angles of any triangle?

B A

C

D 120° 60° 50°

70° Please note: Each exterior angle and its interior angle is supplementary.

Example: ∠CDB + ∠CDA = 180°

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You try:

1. Solve for x and find the measure of the angles Q & M.

2. Find the measure of the vertex ∠ of an isosceles ∆ if either of the exterior angles formed by extending the base measures 144°.

3. Find the measure of a base ∠ of an isosceles ∆ if the exterior angle at the vertex measures 132°

4. In ∆RST, angle S is a right angle and the m∠T = 38°. Find the measure of the exterior angle at R.

P M

153° (3x – 8)°

(2x + 6)° Q

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3.4 Line Segments Associated with the Triangle

There are three types of line segments that exist in the triangle.

1. A ___________________ is a line segment that is drawn from a vertex to the midpoint of the opposite side. A

B M C

In ∆ABC, above, if M is the midpoint of BC, then AM is a median. - -- We can also draw medians to sides AB and AC, once we locate their midpoints.

2. An _____________________ is a line segment drawn perpendicularly from a vertex to the opposite side, forming right angles A

How many altitudes does a triangle have?

In the triangle at the left, AD is an altitude

AD ⊥ BC [Grabyour

[Grabyour

B D C

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3. An ____________________ __________________ in a triangle cuts an angle of the triangle into two congruent angles. A

B D C

** The problem has to say, either directly (using the words “altitude” or “median” or “angle bisector”) or indirectly, by giving you the information that permits you to draw the correct conclusion.

You try: For #’s 1- 6 Describe line segment DF in each of the following triangles.

D D D 1. 2. 3. 4. C C E F E C F E F E _____________ ______________ ______________ _____________ D D 5. 6. C 2 cm E F E _________________ ________________

In the triangle at the left, AD is the angle bisector of angle A. Therefore, ∠BAD ≅ ∠CAD

42° 42°

F 2 cm

D

F

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**There are times when a single line segment can perform two or even all three of these jobs.

In which triangle(s) can this occur?

Illustrate below:

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3.5 Triangle Inequalities

C

H K

The sum of two sides of a triangle is _________________ than the third side. In particular, the sum of the lengths of the two shortest sides must be greater than the longest side.

**Recall that the exterior angle of a triangle is equal to the sum of the non-adjacent interior angles. As a result, that exterior angle must be greater than either non-adjacent interior angle. (i.e. If you have to add two interior angles to get the exterior, then that exterior MUST be greater than either of the two angles you added.)

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3.6 Lengths of Line Segments within Triangles 1) The line segment joining the midpoints of two sides of a triangle is _______________ to the third side and ___________ its length. A

B C In the diagram above, D is the midpoint of AB and E is the midpoint of AC. As a result, DE is parallel to BC, and half its length. 2) The median to the hypotenuse of a right triangle is _____________ the length of the hypotenuse. A

B C

In the diagram above, BD is the median to hypotenuse AC. BD is half the length of AC.

D E

D

*mark diagram

*mark diagram

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You try: 1. Can these sets of numbers be the sides of a triangle? a) {3, 4, 5} b) {11, 6, 9} c) {2, 8, 10} d) {.5, 12, 12} e) {7, 7, 7} f) {13, 30, 13} g) {6¼, 4½, 11} h) {1, 1, 3} i) {15, 8, 17} For # 2 & 3 use this information: In triangle RST, QP joins the midpoints of sides RS and TS, respectively. 2. Find the length of QP if RT is a) 14 b) 17 c) 26.5 d) 6x e) 31½ f) 8¾ 3. Find the length of RT if QP is a) 9 b) 13 c) 10.36 d) 6¾ e) 29½ f) (x+3)

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3.7 Angle - Side Relationship in a Triangle 1) The longest side of a triangle is opposite the triangle’s _______________ angle. Likewise, the largest angle will be opposite the triangle’s _______________ side. Ex: The Right Triangle The hypotenuse of the right triangle is the longest side of this triangle & it is opposite the 90° angle, the largest angle. 2) The shortest side of a triangle is opposite the triangle’s _______________ angle. Likewise, the smallest angle will be opposite the triangle’s ________________ side. You try: 1. In ΔABC, ∠A = 50° & ∠B = 60°. 2. In ΔABC, AB = 11, BC = 10, and Name the longest and shortest AC = 15. Name the largest and sides of ΔABC. smallest angles of ΔABC.

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3.8 The Isosceles and Equilateral Triangles The Isosceles and Equilateral triangles have some very useful properties: ________________ _______________ of an isosceles triangle are congruent. Recall from section 3.1 that the base angles are the angles touching the base. (The other angle is referred to as the _______________ angle.) The Converse (the reverse) of that rule is true as well: If two angles of a triangle are congruent, the sides opposite them are as well. (Note: very often, the converse of a true statement is NOT true; this is one of the rare occasions when both are true.)

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Another special property of the isosceles triangle is that the altitude is also the median is also the bisector of the vertex angle; all three segments fall in the same place. For example, in the diagram below, if we know that AD is an altitude, we know the following:

- AD is perpendicular to BC. (∠ADB and ∠ADC are right angles.) - AD bisects BC (so BD ≅ DC) - AD bisects ∠A ( so ∠BAD ≅ ∠CAD)

A B D C The Equilateral Triangle: - has 3 congruent angles. - every equilateral triangle is also equiangular - each angle of an equilateral triangle measures 60 degrees

** Because an equilateral triangle is also isosceles, all the properties for the isosceles triangle also apply to the equilateral.

*mark diagram

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3.9 The Pythagorean Theorem

The Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Or: In a right triangle, _____ + _____ = _____, where a and b are the legs and c is the hypotenuse. **The a and b values are interchangeable, but c MUST be the hypotenuse. (Remember that the hypotenuse of a right triangle is the side across from the right angle.) **The Pythagorean Theorem can ONLY be used in a right triangle—no other triangles have a hypotenuse. And it can only be used to find the SIDES of a right triangle, never the angles. **When using Pythagorean Theorem, if your answer is an irrational number - the square root of a number that’s not a perfect square - leave your answer in simplest radical form.

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Pythagorean Triples—three whole numbers that work in the Pythagorean Theorem. **Pythagorean Triples are common right triangles. If you have two of the three numbers in a triple, and they’re in the correct positions, you can know the third number without doing the math. **most common Pythagorean Triple is the 3-4-5 triple

32+ 42= 52, so if you’re given a right triangle with legs of 3 and 4, you can simply state that the hypotenuse is 5, because it’s a 3-4-5 triple. - Note, it’s not any 3 consecutive integers that will work; it’s these particular three.) **If you take a triple, and multiply each side by the same amount, you get another triple. Ex: 3-4-5 triple, multiply each side by 2, you get a 6-8-10 triple Check and you’ll see that it, too, works for the Pythagorean Theorem. The most popular triples are, in order:

1. ______________

2. ______________

3. ______________

4. ______________

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**They’re not the only triples that exist - Here’s a list of a few more: http://www.tsm-resources.com/alists/trip.html For a great take on the Pythagorean Theorem and what it DOESN’T say, take a look at what happened when the Scarecrow from the Wizard of Oz was granted a brain: http://www.teachertube.com/video/wizard-of-oz-and-the-pythagorean-theorem-145155 You try: 1. Find the hypotenuse of a right triangle, if the legs are: a) 9 & 12 b) 2 & 3 c) √5 & 6 d) 1.5 & 2 2. Find the other leg when the hypotenuse and one leg is given: a) 26 & 10 b) 8 & 4 c) √17 & 3 d) 50 & 30 3. In an isosceles right triangle, what are the measures of the legs if the hypotenuse is 10?

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3.10 Special Right Triangles The first is derived from the Equilateral Triangle. It is the 30-60-90 degree triangle.

- The shorter leg is _________ the hypotenuse. - The longer leg equals the shorter leg times ____________.

**The short leg is opposite the ________ degree angle. **The long leg is opposite the __________. **The hypotenuse is opposite the ___________ angle.

x√3 2x Examples:

***We will also investigate what happens when the side opposite the 60 degree angle is whole number.***

60°

x

10

5

5√3

60°

30°

30°

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The second is the _________________ ______________ Triangle, or the 45-45-90 degree triangle.

- The legs are _________________. - The hypotenuse equals the leg times ______________.

When the hypotenuse of the Isosceles Right Triangle is a whole number then:

- A leg equals the hypotenuse times √2/2. ***However, if you forget these rules for the Isosceles Right Triangle, you can always use the Pythagorean Theorem to find the lengths of the legs or hypotenuse.

x x√2

x x√2 2

x√2 2

x

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You try: (remember to draw pictures for each when solving) 30-60-90 triangle 1. Find the remaining two sides when the hypotenuse is 12. 2. Find the remaining two sides when the side opposite the 30 degree angle is 7. 3. Find the remaining two sides when the side opposite the 60 degree angle is 9√3. 4. Find the remaining two sides when the hypotenuse is 8√3.*

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You try: (remember to draw pictures for each when solving) 45-45-90 triangle 1. Find the remaining two sides when the hypotenuse is 8√2. 2. Find the remaining two sides when one leg is 6. 3. Find the remaining two sides when the hypotenuse is 14. 4. Find the remaining two sides when one leg is 10√2.