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Triangle Basics

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Definition

• A triangle is a three-sided figure formed by joining three line segments together at their endpoints.

• A triangle has three sides.

• A triangle has three vertices (plural of vertex).

• A triangle has three angles.

1

2

3

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Naming a Triangle

• Consider the triangle shown whose vertices are the points A, B, and C.

• We name this triangle by writing a triangle symbol followed by the names of the three vertices (in any order).

A B

C

∆Name

ABC

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The Angles of a Triangle

• The sum of the measures of the three angles of any triangle is• Let’s see why this is true.• Given a triangle, draw a line through one of its vertices parallel to the

opposite side.• Note that because these angles form a

straight angle.• Also notice that angles 1 and 4 have the same measure because

they are alternate interior angles and the same goes for angles 2 and 5.

• So, replacing angle 1 for angle 4 and angle 2 for angle 5 gives

4 3 5 180m m m∠ + ∠ + ∠ = °

1 2

34 5

1 3 2 180 .m m m∠ + ∠ + ∠ = °

180 .°

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Example

• In

• What is

, 95 and 37 .DEF m D m E∆ ∠ = ° ∠ = °?m F∠

180m D m E m F∠ + ∠ + ∠ = °95 37 180m F° + ° + ∠ = °

132 180m F° + ∠ = °180 132m F∠ = ° − °48m F∠ = °

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Example

• In the figure, is a right angle and

bisects

• If then what is

A B C

D

C∠DB .ADC∠

65 ,m DBC∠ = ° ?m A∠

9065

25

40

In , 180 (65 90 ) 25 .BCD m BDC∆ ∠ = ° − ° + ° = °Since is a bisector, 2(25 ) 50 .DB m ADC∠ = ° = °

50

In , 180 (90 50 ) 40 .ACD m A∆ ∠ = ° − ° + ° = °

?

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Angles of a Right Triangle

• Suppose is a right triangle with a right angle at C.

• Then angles A and B are complementary.

• The reason for this is that

A

B

C

ABC∆

180m A m B m C∠ + ∠ + ∠ = °90 180m A m B∠ + ∠ + ° = °

90m A m B∠ + ∠ = °

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Exterior Angles

• An exterior angle of a triangle is an angle, such as angle 1 in the figure, that is formed by a side of the triangle and an extension of a side.

• Note that the measure of the exterior angle 1 is the sum of the measures of the two remote interior angles 3 and 4. To see why this is true, note that

1 2 3

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1 180 2 (because of the straight angle)m m∠ = ° − ∠3 4 180 2 (because of the triangle)m m m∠ + ∠ = ° − ∠So, 1 3 4m m m∠ = ∠ + ∠

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Classifying Triangles by Angles

• An acute triangle is a triangle with three acute angles.

• A right triangle is a triangle with one right angle.

• An obtuse triangle is a triangle with one obtuse angle.

acute triangle right triangle obtuse triangle

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Right Triangles

• In a right triangle, we often mark the right angle as in the figure.

• The side opposite the right angle is called the hypotenuse.

• The other two sides are called the legs.

B C

A

hypo

tenu

se

leg

leg

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Classifying Triangles by Sides

• A triangle with three congruent sides is called equilateral.

• A triangle with two congruent sides is called isosceles.

• A triangle with no congruent sides is called scalene.

equilateral isosceles scalene

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Angles and Sides

• If two sides of a triangle are congruent…• then the two angles opposite them are

congruent.• If two angles of a triangle are congruent…• then the two sides opposite them are

congruent.

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Equilateral Triangles

• Since all three sides of an equilateral triangle are congruent, all three angles must be congruent too.

• If we let represent the measure of each angle, then

x

180

3 180

60

x x x

x

x

+ + = °= °

= °

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Isosceles Triangles

• Suppose is isosceles where

• Then, A is called the vertex of the isosceles triangle, and is called the base.

• The congruent angles B and C are called the base angles and angle A is called the vertex angle.

ABC∆.AB AC≅

BC

A

B

C

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Example

• is isosceles with base

• If is twice then what is

• Let denote the measure of

• Then

A

B C

ABC∆.BC

m B∠ ,m A∠?m A∠

x.A∠

x

2x 2x2 .m B x m C∠ = = ∠

Then 2 2 5 180 .x x x x+ + = = °So, 180 / 5 36 .m A x∠ = = ° = °

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Example

• In the figure,

and

• Find

• Since is isosceles,

the base angles are congruent. So,

A

B

C

D

,AD BD≅20 ,m C∠ = ° 25 .m A∠ = °

and .m CBD m CDB∠ ∠

20

25

25

110

130

50ABD∆

25 .m ABD∠ = °

Then 130 since the angles of

must add up to 180 .

m ADB

ADB

∠ = °∆ °Then 50 since and

are supplementary.

m BDC ADB

BDC

∠ = ° ∠∠Then 110 since the angles of

must add up to 180 .

m CBD

BCD

∠ = °∆ °

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Inequalities in a Triangle

• In any triangle, if one angle is smaller than another, then the side opposite the smaller angle is shorter than the side opposite the larger angle.

• Also, in any triangle, if one side is shorter than another, then the angle opposite the shorter side is smaller than the angle opposite the longer side.

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Example

• Rank the sides of the triangle below from smallest to largest.

• First note that

• So,

A B

C

55° 53°

180 (55 53 ) 72 .m C∠ = ° − ° + ° = °

72°

.AC BC AB< <

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Medians

• A median in a triangle is a line segment drawn from a vertex to the midpoint of the opposite side.

• An amazing fact about the three medians in a triangle is that they

all intersect in a common point. We call this point the centroid of the triangle.

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• Another fact about medians is that the distance along a median from the vertex to the centroid is twice the distance from the centroid to the midpoint.

2x

x

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Example

• In the medians are drawn, and the centroid is point G.

• Suppose

• Find

A

B

CG

M

N

P

ABC∆

12,AM = 7,BM =and 9.BG =

, , and .GM MC GN

4 7

4.5

Let and 2 . Then

2 12, and so 4.

GM x AG x

x x GM x

= =+ = = =

Since is a midpoint,

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M

MC BM= =Since is half of ,

0.5(9) 4.5

GN BG

GN = =

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Midlines

• A midline in a triangle is a line segment connecting the midpoints of two sides.

• There are two important facts about a midline to remember:

midline

A midline is parallel to one side

of the triangle.

A midline is half the length of

the side to which it is parallel.

x

2x

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Example

• In D and E are the midpoints of respectively.

• If and then find and

BA

C

D E

,ABC∆ and , AC BC

7.5DE = 56m ABC∠ = °AB .m BED∠

is a midline. So,

2 2(7.5) 15.

DE

AB DE= = =

and and are

interior angles on the same side of the

transversal. So, they are supplementary.

So, 180 56 124 .

AB DE ABC BED

m BED

∠ ∠

∠ = ° − ° = °

P

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The Pythagorean Theorem

• Suppose is a right triangle with right angle at C.

• The Pythagorean Theorem states that

• Here’s another way to state the theorem: label the lengths of the sides as shown. Then

ABC∆

B C

A

2 2 2BC AC AB+ =

a

bc

2 2 2a b c+ =

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• In words, the Pythagorean Theorem states that the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse, or:

2 2 2leg leg hypotenuse+ =le

g

leg

hypotenuse

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Example

• Suppose is a right triangle with right angle at C.

B C

A

ABC∆

If 5 and 12 find .BC AC AB= =

2 2 2

2

2

5 12

25 144

169

169 13

AB

AB

AB

AB

+ =+ ==

= =

If 5 and 5 find .BC AC AB= =

2 2 2

2

2

5 5

25 25

50

50 25 2 5 2

AB

AB

AB

AB

+ =+ ==

= = =g

If 12 and 6 find .AB BC AC= =

2 2 2

2

2

6 12

36 144

144 36 108

108 36 3 6 3

AC

AC

AC

AC

+ =+ =

= − =

= = =g

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45-45-90 Triangles

• A 45-45-90 triangle is a triangle whose angles measure

• It is a right triangle and it is isosceles.

• If the legs measure then the hypotenuse measures

• This ratio of the sides is memorized, and if one side of a 45-45-90 triangle is known, then the other two can be obtained from this memorized ratio.

45 , 45 , and 90 .° ° °

x2.x

x

x

2x

45

45

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Example

• In is a right angle and• If then find• First notice that too since

the angles must add up to • Then this is a 45-45-90 triangle and

so:

45A C

B

,ABC∆ C∠ 45 .m A∠ = °6,AB = .BC

6 ?

45m B∠ = °180 .°

: : : : 2.

So, 2 6 and

6 6 2 6 23 2

22 2 2

So, 3 2.

AC BC AB x x x

x

x

BC

=

=

= = = =

=

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30-60-90 Triangles

• A 30-60-90 triangle is one in which the angles measure

• The ratio of the sides is always as given in the figure, which means:

• The side opposite the angle is half the length of the hypotenuse.

• The side opposite the angle is times the side opposite the angle.

A B

C

60°

30°

x

2x 3x

30 , 60 , and 90 .° ° °

30°

60°3 30°

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Example

• In • If find • First note that, since the three angles

must add up to

• So this is a 30-60-90 triangle.

A C

B

60°

30°

, 60 and 30 .ABC m A m B∆ ∠ = ° ∠ = °12,BC = .AB

180 ,° 90 .m C∠ = °

: : 2 : : 3

So, 12 3, which gives

12 12 3 12 34 3.

33 3 3

So, 2 8 3

AB AC BC x x x

x

x

AB x

=

=

= = = =

= =

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The Converse of the Pythagorean Theorem

• Suppose is any triangle where

• Then this triangle is a right triangle with a right angle at C.

• In other words, if the sides of a triangle measure a, b, and c, and

then the triangle is a right triangle where the hypotenuse measures c.

ABC∆2 2 2.AC BC AC+ =

2 2 2a b c+ =

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Example

• Show that the triangle in the figure with side measures as shown is a right triangle.

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2425

2 2 249 5767 24 25 562+ = =+ =

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If Yes ,Join Dreams School “Campaign for Female Education”

Help us in bringing a change in a girl life, because “When someone takes away your pens you realize quite how important education is”.

Just Click on any advertisement on the page, your one click can make her smile. We our doing our part & u ?Eliminate Inequality “Not Women”