OTCQ

RGT is adjacent to angle

TGL.

If m RGL = 100 ◦ and m TGL = 20

◦,

What is m RGT?

OTCQ

RGT is adjacent to angle

TGL.

If m RGL = 100 ◦ and m TGL = 20

◦,

What is m RGT?R

20 ◦

80 ◦

L

T

G

Aims 1-6 & 1-7 How do we define angles and triangles?

NY GG 31, GG 32, GG 33, GG 35, GG 28

Read 1-6 and 1-7 do problems 18 -25 on page 22.

1-6 & 1-7 objectives

1. SWBAT name angles and angle parts.

2. SWBAT define:

congruent angles, complimentary angles, supplementary angles and adjacent angles

Bisector of an angle

Perpendicular line

Distance from a point to a line

3. SWBAT add angles.

4. SWBAT define a triangle and its parts.

• An angle consists of two different rays that have the same initial point.

• The rays are the sides of the angle.• The initial point A is the vertex of the angle. • The angle that has rays AB and AC as

sides may be named BAC, CAB, or A.

C

B

Vertex A

sides

The set of all points between the sides of the angle is the interior of an angle.

The exterior of an angle is the set of all points outside the angle.

Angle NameR, SRT, TRS, or 1

If the point is the vertex of more than one angle, you must use all three points to name the angle. The middle point is always the vertex.

The measure of A is denoted by mA.

The measure of an angle can be approximated using a protractor, using units called degrees(°).

BAC has a measure of 50°, which can be written as mBAC = 50°.

B

A

C

Angle RelationshipsCongruent Angles: Angles with equal measures in

degrees.

MSU EWR or USM RWEM

R

E

W

S U

Angle Relationships• Complementary Angles: Two angles

are called complementary angles if the sum of their degree measurements equals 90° degrees.

• Supplementary Angles: Two angles are called supplementary angles if the sum of their degree measurements equals 180° degrees.

Angle Relationships• Adjacent Angles: Share a

vertex and a common side but no interior points.

• Bisector of an angle: a ray that divides the angle into two congruent angles.

is the angle bisector.OY

What are perpendicular lines?

• Two lines that

intersect at a 90 ◦

angle (right angle) are Perpendicular lines.

• Boxed vertex

means 90 ◦

The shortest distance from a point to a line will always be the distance of the perpendicular line from the point to the line.

Boxedvertexmeans

90 ◦

Adding Angles

When you want to add angles, use the notation m1, meaning the measure of 1.

If you add m1 + m2, what is your result?

m1 + m2 = 58.

22°

36°

21

D

B

C

A

Therefore, mADC = 58.

m1 + m2 = mADC also.

Congruent Congruent

Angles that measure the same in degrees are congruent.

PET TEJ

LEP JEL

Symbol for congruentSymbol for congruent

EL

P

T

J

Congruent Congruent

Explain why the angles are congruent.

PET TEJ

LEP JEL

Symbol for congruentSymbol for congruent

EL

P

T

J

Congruent Congruent

They are all 90◦

PET TEJ

LEP JEL

Symbol for congruentSymbol for congruent

EL

P

T

J

Congruent Congruent

Segments that measure the same in inches, feet, miles, milimeters, centimeters, meters, or kilometers are called congruent.

___ ___

AT PC

Symbol for congruentSymbol for congruent

A PT

C

1-6 & 1-7 objectives CHECK UP

1. SWBAT name angles and angle parts.

2. SWBAT define:

congruent angles, complimentary angles, supplementary angles and adjacent angles

Bisector of an angle

Perpendicular line

Distance from a point to a line

3. SWBAT add angles.

4. NEXT:SWBAT define a triangle and its parts.

Polygon: A closed plane figure formed by three or more line segments that meet only at their endpoints.

Triangle: a polygon with exactly 3 sides.

Equilateral all 3 sides and angles are congruent. Also called Equiangular.

Isosceles Triangle— 2 or 3 sides/angles are congruent

Scalene—no sides or angles are congruent

Types of Triangles:

Triangle Parts:• 3 vertices: A, B, C.• 3 line segments:

__ __ __BC , CA & AB

• Any 2 line segments in a triangle that share a common endpoint are called adjacent sides. B

C

A

__

CB is the opposite side of endpoint A

__ __CA and BA share endpoint A, so they are adjacent sides. __ __CB and BA share endpoint B, so they are adjacent sides. __ __BC and CA share endpoint C, so they are adjacent sides.

EQUILATERAL is EQUIANGULAR

• 3 congruent 60◦ acute angles (“equiangular”).

• 3 congruent line segments. (“equilateral” ).• Acute. • Can’t be right

• Also isosceles

because at least 2 sides/angles are congruent.• Each line segment is a side.

• The 2 congruent sides are called legs. Green dashes means congruent.

• The noncongruent side is called the base. No green dashes.

• The 2 congruent angles are called base angles. Blue arc means congruent.

• The noncongruent angle is called the vertex angle. No blue arcs.

legleg

Base

Base Base

Angle Angle

Isosceles Triangles need only 2 congruent sides/angles.

Vertex Angle

ISOCELES ISOCELES Right TriangleRight Triangle

• A right triangle with 2 congruent sides/angles, is an isosceles right triangle.

• its base angles are each

45◦• The red side is both a hypotenuse and a base.

• The boxed 90◦ right angle is the vertex angle.

• Green dashes mean congruent. • Blue arcs mean congruent.

45◦ base

Hypotenuse angle

base

leg

leg

45 ◦ 90◦ Base angle vertex

Acute Triangle: an acute triangle has 3 acute angles.

Could be equilateral/equiangular (all = 60◦).Could be isosceles ∆ or scalene ∆ but never a right and never an obtuse ∆.

mCAB = 41.76 mBCA = 67.97

mABC = 70.26

B

A

C

Right Triangle

• 1 right angle

Obtuse Triangle: has one obtuse angle.

Right and Obtuse triangles are never acute or equilateral, but can be isosceles (have 2 congruent sides/angles).

Right Triangle

• Red represents the hypotenuse of a right triangle.

• The blue sides that form the right angle are the legs.

hypotenuseleg

leg

leg

leg

base

What is this?

leg

leg

base

What is this? An isosceles triangle.

Quick Algebra Review

Commutative PropertyCommutative Property of Addition: a + b = b + a

Commutative Property of Multiplication: ab = ba

Examples

2 + 3 = 5 = 3 + 2

3• 4 = 12 = 4 • 3

The commutative property does not work for subtraction or division!!!!!!!!

Associative PropertyAssociative property of Addition:

(a + b) + c = a + (b + c)

Associative Property of Multiplication:

(ab) c = a (bc)

Examples

(1 + 2) + 3 = 1 + (2 + 3)

(2 • 3) • 4 = 2 • (3 • 4)

The associative property does not work for subtraction or division!!!!!

Identity Properties

1) Additive Identity

a + 0 = a

2) Multiplicative Identity

a • 1 = a

Inverse Properties

1) Additive Inverse (Opposite)

a + (-a) = 0

2) Multiplicative Inverse (Reciprocal)

a 1

a 1

Multiplicative Property of Zero

a • 0 = 0

(If you multiply by 0, the answer is 0.)

The Distributive PropertyAny factor outside of expression enclosed within

grouping symbols, must be multiplied by each term inside the grouping symbols.

Outside left or Outside right

a(b + c) = ab + ac (b + c)a = ba + ca

a(b - c) = ab – ac (b - c)a = ba - ca

Time permitting start homework