december 2, 2015 5.1 angles of triangles - mesa public … measure of an exterior angle of a...

5.1 Angles of TrianglesDecember 2, 2015

Geometry

5.1 Angles of Triangles

Essential Question

How are the angle measures of a triangle

related?



Goals – Day 1

Classify triangles by their sides

Classify triangles by their angles

Identify parts of triangles.

Find angle measures in triangles.


Triangle Symbol

Use the picture for triangle.


Triangle

A triangle is a figure formed by three

segments joining three noncollinear points.

A

B

C

This is ABC, which can also be named BCA,

CAB, BAC, CBA, or ACB.


Classifying Triangles by Sides

Equilateral

Isosceles

Scalene


Equilateral Triangle

Three congruent sides.


Isosceles Triangle

At least two congruent sides.


Scalene Triangle

No congruent sides.


Classifying Triangles by Angles

Acute

Equiangular

Right

Obtuse


Acute Triangle

Three acute angles

Three Congruent Angles


Equiangular Triangle


One Right Angle

December 2, 2015

Right Triangle


Obtuse Triangle

One Obtuse Angle


And to add to the confusion…

An equilateral triangle is also equiangular.

An equiangular triangle is also acute.

An equilateral can be considered an

isosceles triangle.

An equilateral triangle is also acute.


Vertex

Each of the three points joining the sides

of a triangle is a vertex.

There are three vertices in each triangle.

Points A, B, and C are the vertices.

A

B

C


Adjacent Sides

Two sides that share a common vertex are

adjacent sides.

The third side is the opposite side.

R T

AIn RAT, RA and RT are

adjacent sides.

AT is the opposite side from

∠𝑅.


Isosceles Triangles (In this case, we consider an isosceles

triangle with only two congruent sides.)

The congruent sides are the LEGS.

The third side is the BASE.

Leg Leg

Base


Right Triangle

The LEGS form the right angle.

The third side (opposite the right angle) is

the Hypotenuse.

Leg

Leg


Hypotenuse

From the Greek “stretched against”.

Always longer than either leg.


What have you learned so far?

In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅𝑀𝑄. Complete the following sentence.

1. Name the legs of the

isosceles triangle PMQ.

Segments PM and QM.

P

Q

N M




2. Name the base of

isosceles triangle PMQ.

Segment PQ.

P

Q

N M




3. Name the hypotenuse of

right triangle PNM.

Segment PM.

P

Q

N M




4. Name the legs of right

triangle PNM.

Segments NP and NM.

P

Q

N M




5. Name the acute angles

of right triangle QNM.

Q and NMQ

P

Q

N M


Example 1

Classify these triangles by its angles and by

its sides.

a. c. b.

125°

Right , Scalene

Obtuse ,

Isosceles

Equiangular, Equilateral

Isosceles , Acute


Example 2

Complete the sentence with always,

sometimes, or never.

a. An isosceles triangle is ________ a right

triangle.

b. An obtuse triangle is ________ a right triangle.

c. A right triangle is ________ an equilateral

triangle.

d. A right triangle is ________ an isosceles

triangle.

Sometimes

Never

Never

Sometimes


Important Triangle Theorems

5.1 Triangle Sum Theorem

5.2 Exterior Angle Theorem


5.1 Triangle Sum Theorem

The sum of the measures of the interior

angles of a triangle is 180°.

A

B

C

mA + mB + mC = 180°

A Proof of the Triangle Sum Thm

1. Given

2. Draw line through point B parallel to AC

2. Parallel Postulate (3.1)

3. m4 + m3 + m5 = 180 3. Def. of Straight Angle

4. Alternate Interior ’s

1. ABC

4. m1 = m4 and m2 = m5

Statements Reasons

Given: ABC

Prove: m1 + m2 + m3 = 180°

December 2, 2015 5.1 Angles of Triangles

1 2

34 5

A

B

C

5. m1 + m3 + m2 = 180 5. Substitution


Example 3

Find the measure of 1.

Solution:

m1 + 70 + 32 = 180

m1 + 102 = 180

m1 = 180 – 102

m1 = 78°

70° 32°

1


Example 4In MAD:

mM = (2x)°

mA = (3x)°

mD = (4x)

Find the measure of each angle, and classify.

Solution:

2x + 3x + 4x = 180

9x = 180

x = 20

= 2(20) = 40°

= 3(20) = 60°

= 4(20) = 80°

This triangle is acute.


Example 5

In RST:

mR=(5x + 10)

mS=(2x + 15)

mT=(3x + 35)

Find the measure of the three angles and

then classify the triangle by angles.


Example 5 Solution

(5x + 10) + (2x + 15) + (3x + 35) = 180

10x + 60 = 180

10x = 120

x = 12

mR=(5x + 10) = 5(12) + 10 = 70

mS=(2x + 15) = 2(12) + 15 = 39

mT=(3x + 35) = 3(12) + 35 = 71

ACUTE


Your Turn

In ABC:

mA=(x + 30)

mB=x

mC=(x + 60)




Your Turn Solution

RIGHT

𝑥 + 30 + 𝑥 + 𝑥 + 60 = 180

3𝑥 + 90 = 180

3𝑥 = 90

x = 30

m∠𝐴 = 30 + 30 = 60°

m∠𝐵 = 30°m∠𝐶 = 30 + 60 = 90°

In ABC:

mA=(x + 30)

mB=x

mC=(x + 60).


Your Turn Again.

In ABC:

mA=(6x + 11)

mB=(3x + 2)

mC=(5x - 1)




Your Turn Again Solution

ACUTE

6𝑥 + 11 + (3𝑥 + 2) + 5𝑥 − 1 = 180

14𝑥 + 12 = 180

14𝑥 = 168

x = 12

m∠𝐴 = 6(12) + 11 = 83°

m∠𝐵 = 3 12 + 2 = 38°m∠𝐶 = 5 12 − 1 = 59°

In ABC:

mA=(6x + 11)

mB=(3x + 2)

mC=(5x - 1).


Assignment

Geometry


Essential Question

How are the angle measures of a triangle

related?



5.1 Day 2

Yesterday:

The Interior Angle Theorem: the sum of

the interior angles of a triangle is 180°.

Today:

The Exterior Angle Theorem


But First…

A corollary to the interior angle theorem.

A corollary is a theorem that can be

proved easily from another theorem.

Not “big” enough to warrant title of

theorem.

A corollary follows from a theorem.


Corollary to Theorem 5.1

The acute angles of a right triangle are

complementary.

1

2

m1 + m2 + 90 = 180

m1 + m2 = 90

QED


Example 1Find X

20°

x°

x = 70°

Since this is a right triangle, the

acute angles are complementary,

and 90 – 20 = 70.


Interior and Exterior Angles

Start with a

triangle…


Extend the

sides….

1

2

3

1, 2, 3 are INTERIOR ANGLES.

They are INSIDE the triangle.


1

2

3

4, 6, 8, 9, 10, and 12 are

EXTERIOR ANGLES.

They are OUTSIDE the triangle.

They are ADJACENT to the interior

angles.

4

6

8 9

10

12


1

2

3

5, 7, and 11 are NOT EXTERIOR

ANGLES.

They are simply vertical angles to the

interior angles.

5

7

11


It is common (and less confusing) to draw

only one exterior angle at a vertex.

1 2

3

4

5

6

Interior Angles: 1, 2, 3

Exterior Angles: 4, 5, 6

Exterior angles are always supplementary to the interior angles.


5.2 Exterior Angle Theorem

The measure of an exterior angle of a

triangle is equal to the sum of the measures

of the two nonadjacent interior angles.

12

3

m1 = m2 + m3


Note:

Sometimes (usually) the two nonadjacent

interior angles are referred to as REMOTE

INTERIOR ANGLES. The theorem then

reads:

An exterior angle of a triangle is equal to

the sum of the two remote interior angles.


5.2 Exterior Angle Thm Proof (Informal)

12

3

4

m2 + m3 + m4 = 180 ( angle sum)

m4 + m1 = 180 (linear pair postulate)

m2 + m3 + m4 = m4 + m1 (substitution)

m2 + m3 = m1 (subtraction)


Naming Remote Interior Angles

1 2

3

46

5

8 7

9

For exterior 1, the remote

interior angles

are_____________.6 & 8



1 2

3

46

5

8 7

9


interior angles

are_____________.2 & 8



1 2

3

46

5

8 7

9


interior angles

are_____________.2 & 6



1 2

3

46

5

8 7

9

For remote interior angles

6 & 8, the exterior angle

is _____________.1 or 3



1 2

3

46

5

8 7

9



is _____________.7 or 9



1 2

3

46

5

8 7

9



is _____________.4 or 5


Example 2

110°1

45°

Find m1.

By Theorem 5.2:

m1 + 45 = 110

m1 = 110 – 45 = 65°


Example 3

(x + 15)° (3x – 10)°

45°

Solve for x.

(x + 15) + 45 = 3x – 10

x + 60 = 3x – 10

70 = 2x

x = 35


Problems for You

Use the exterior angle theorem!

Write down the equation for each problem

and solve.


Your Turn.

1. Find m1

1125

32

Solution:

m1 = 32 + 125

m1 = 157


2. Find m2

2

45

Solution:

m2 + 45 = 165

m2 = 120

165


3. Solve for x.

(2x + 30)° 60

110°

Solution:

2x + 30 + 60 = 110

2x + 90 = 110

2x = 20

x = 10


4. Solve for x.

(5x) (12x – 4)

(6x + 8)

Solution:

12x – 4 = (6x + 8) + 5x

12x – 4 = 11x + 8

x = 12


5. Solve for x.

(3x + 2)

(5x – 10)

(7x + 3)

Solution:

(3x + 2) + (5x – 10) = 7x + 3

8x – 8 = 7x + 3

x = 11


A Final Challenge Problem…

Find the measure of each numbered angle.

40°

30°

60°

20°

1

2 3

4 5

6 7

50°

90°

60°

60° 60°

60°

100°


Summary

The sum of the interior angles of a triangle

is 180 degrees.

The acute angles of a right triangle are

complementary.

An exterior angle is equal to the sum of

the two remote interior angles.


Assignment

december 2, 2015 5.1 angles of triangles - mesa public … measure of an exterior angle of a...

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