Angles

Ch 3

Angles

3.1

Opposite Rays Opposite Rays –Rays that are part of the same line and have only their

endpoint in common

Also called straight angle

XZXY and

Angle Another case when two rays share an endpoint… Angle –A figure formed by two noncollinear rays that have a

common endpoint Vertex – common endpoint Sides – rays that make up angle

Three ways to name: Three points (vertex in middle)

Vertex only

A number

Example Name the angle in four ways. Then identify its

vertex and its sides.

Naming Angles If more than one angle share a vertex, name the

angle with three points or a number

Example Name all angles having D as their vertex.

Interior/Exterior Angle separates a plane into three regions Interior of the angle Exterior of the angleAngle itself

Example Tell whether each point is in the interior,

exterior, or on the angle.

Assignment P92: 1 – 32 (omit 30) 21 – 23: if false, tell why

Read P95, #1 – 3

Angle Measure

3-2

Angle Measure Degree –Unit that angles are measured in 1/360th of a circle Symbol: °

reads ‘the measure of angle PQR’ Reason why there is no degree sign, because it is a measure

and not a measurement

PQRm

Postulate 3-1: Angles Measure Postulate

Protractor Protractor –Geometric tool used to measure angles and sketch

angles of given measure

Example Use a protractor to measure angle KLM.

Examples Use a protractor to

measure ∠CDF. Find the measure of

∠PQR, ∠PQS, and ∠PQT.

Postulate 3-2: Protractor Postulate

Meaning: from a given ray, there are two ways to draw an angle with a given angle measure

Example Use a protractor to draw an angle having a

measure of 35.

Use a protractor to draw an angle having a measure of 65.

Classifying Angles Types of Angles:Right Angle – measure is 90

Acute Angle – measure is less than 90

Obtuse Angle – measure is more than 90

Example Classify each angle as acute, obtuse, or right

Example The measure of angle A is 100. Solve for x.

The measure of angle B is 138. Solve for x.

Assignment P100: 1, 2 (use P99), 4 – 25, 31 – 37 3.1/3.2 Wkst

Ch 3 Investigation Those Magical Midpoint P102: 1 – 2

The Angle Addition Postulate

3-3

Hands-On Geometry P104: Follow steps, answer questions

Postulate 3-3: Angle Addition Postulate (AA Postulate)

Example

Example

Example Find m∠ABC if

m∠ABD = 70 and m∠DBC = 43.

If m∠EBC = 55 and m∠EBD = 20, find m∠2.

Find m∠ABD if m∠ABC = 110 and m∠2 = 36.

Angle Bisector

Example

Example

Hands-On Geometry P107: follow steps, answer questions

Assignment P108: 1 – 24, 26 – 30

ADJACENT ANGLES AND LINEAR PAIRS OF ANGLES

3-4

Adjacent Angles

Example

Linear Pair

Name the angle that forms a linear pair with ∠TCM.

Do ∠1 and ∠TCE form a linear pair? Justify your answer.

Example

Example The John Hancock Center in Chicago, Illinois,

contains many types of angles. Describe the highlighted angles.

Assignment P112: 1, 3 – 21, 23 – 27 Read P115 1 – 3

Complementary and Supplementary Angles

3-5

Complementary Angles

Complementary Angles

Supplementary Angles

Supplementary Angles

Example Name two pairs of complementary angles.

Name a pair of adjacent angles.

Example Name a pair of nonadjacent supplementary

angles.

Find the measure of an angle that is supplementary to angle BGC.

Example Angles C and D are supplementary. If m∠C =

12x and m∠D = 4(x + 5), find x. Then find m∠C and m∠D.

Postulate 3-4: Supplement Postulate

Example If m∠3 = 115 and ∠3 and ∠4 form a linear pair,

find m∠4.

Assignment P119: 1 – 37

Congruent Angles

3-6

Congruent Angles

Vertical Angles

Theorem 3-1: Vertical Angle Theorem

Example Find x.

Theorems

Theorems

Example Suppose ∠A ≅ ∠B

and m∠B = 47. Find the measure of an angle that is supplementary to ∠A.

In the figure below, ∠1 is supplementary to ∠2, ∠3 is supplementary to ∠2, and m2∠ = 105. Find m∠1 and m∠3.

Example

Theorems

Assignment P125: 1 (with compass), 4 – 22, 24 – 28

Perpendicular Lines

3-7

Perpendicular Lines

Looking at a Proof

Theorem 3-8

Example

Example

Hands-On Geometry P130 Follow stepsAnswer Questions

Theorem 3-9 Can there be another perpendicular to a line

through a certain point?

Assignment P131: 1, 3 – 27, 29 – 33

Review P134: 1 – 39 P137: 1 – 20

String Art Project