OPENING ACTIVITY GEOMETRY JEOPARDY GEOMETRY JEOPARDY.

OPENING ACTIVITYOPENING ACTIVITY

GEOMETRY JEOPARDY

A REVIEW ABOUT A REVIEW ABOUT COMPLEMENTARY COMPLEMENTARY

ANGLES & ANGLES & SUPPLEMENTARY SUPPLEMENTARY

ANGLESANGLES

Topic: Angle PairsTopic: Angle Pairs

What are What are complementary angles? angles?

What are What are supplementary angles?angles?

Consider the following:Consider the following:

Complementary Complementary AnglesAngles

--Are two angles that Are two angles that together make a together make a right angle. right angle.

The measures of The measures of the two angles the two angles must add up to must add up to 90°. 90°.

A

B C

D

60º

30º

m ABC = m ABD + m CBD

90 = 30 + 60 ABD and CBD are

COMPLEMENTARY ANGLES


Two 45Two 45º angle are º angle are Complementary.Complementary.

RPD and RPD and QPD areQPD are

Complementary angles.Complementary angles.

R

P Q

D

45º

45º

Consider this figureConsider this figure

B and B and Q are Q are Complementary Complementary angles.angles.

B is a complement to B is a complement to Q.Q. QQ is a complement to is a complement to B B..

B Q70º 20º

Can we say that Can we say that B and B and Q are Q are Complementary angles?Complementary angles?

Supplementary AnglesSupplementary AnglesAre two angles that together Are two angles that together

form one-half of a complete form one-half of a complete rotation—that is, 180°.rotation—that is, 180°.

The measures of two The measures of two supplementary angles, supplementary angles, therefore, must add up to 180 therefore, must add up to 180 when added together. when added together.

The supplementary angle of a The supplementary angle of a 50° angle, for example, is a 50° angle, for example, is a 130°.130°.


mm RPD + m RPD + m QPD = 180.QPD = 180. Therefore, Therefore, RPD and RPD and QPD are QPD are

supplementary angles.supplementary angles.

R P Q

D

35º145º

What can you say about the angle sum What can you say about the angle sum measure of measure of RPD and RPD and QPD ? QPD ?

Another illustration:Another illustration:

R and R and P are supplementary P are supplementary angles.angles.

R is a supplement to R is a supplement to P . P .

P is a supplement to P is a supplement to R. R.

R P30º

150º

mm R + m R + m P = 180. P = 180.

Chapter 2: Chapter 2: Angle RelationshipsAngle Relationships

Topic: Angle Pairs Topic: Angle Pairs (continuation)(continuation)

ObjectivesObjectives

Enumerate the different kinds of Enumerate the different kinds of angle pairsangle pairs

Define each kind of angle pair;Define each kind of angle pair;

Identify & Illustrate the different Identify & Illustrate the different kinds of angle pairs, clearly kinds of angle pairs, clearly showing their relations.showing their relations.

IntroductionIntroductionRelationships exist between angles.Relationships exist between angles.

If two angles have the If two angles have the same same measuremeasure, then they are , then they are CONGRUENTCONGRUENT. .

For example, if For example, if mmA = 50A = 50 and and mmB B = 50= 50, then , then A A B .B .

By the sum of their measures, By the sum of their measures, relations can be established.relations can be established.

IntroductionIntroduction

Relationship is very Relationship is very important.important.

- relationship to your fellow - relationship to your fellow students.students.

- relationship to our friends, - relationship to our friends, neighbors, & other people. neighbors, & other people.

IntroductionIntroduction

Relationship to our Relationship to our family.family.

Relationship to God.Relationship to God.

We MUST learn to We MUST learn to value relationships.value relationships.

As the saying goes,As the saying goes,

“ “ No man is an No man is an island. No man can island. No man can

stand alone”stand alone”

Look at this figure…Look at this figure…

Consider Consider RPDRPD and and QPD. QPD.

- - share a common vertexshare a common vertex(P),(P),- Share a common side Share a common side

((segment PDsegment PD) ) - but no interior points in but no interior points in

common.common.

RPD and RPD and QPD are QPD are

Adjacent angles.Adjacent angles.

R

P Q

D

. S

. A

ADJACENT ANGLESADJACENT ANGLES

Are angles meeting at a Are angles meeting at a common vertex common vertex (corner) (corner) and and sharing a common sharing a common side side but but NO interior NO interior points in commonpoints in common..


RPD and RPD and QPD QPD are are Adjacent Adjacent anglesangles & & complementarycomplementary..

R

P Q

D

. S

. A

How about the other pairs of angles in How about the other pairs of angles in the figure?the figure?

Like ,Like , RPD and RPD and QPR ?QPR ? QPD and QPD and QPR ? QPR ?

Are these pairs of angles Are these pairs of angles

Adjacent or not ? why?Adjacent or not ? why?

These pairs of angle are These pairs of angle are NON – ADJACENT NON – ADJACENT

ANGLES.ANGLES.

R

P Q

D

. S

. A


Can we say that Can we say that B and B and Q Q are are Complementary?Complementary?

Adjacent or non-adjacent?Adjacent or non-adjacent?

B and B and Q Q are are Complementary angles BUT Complementary angles BUT non- adjacent anglesnon- adjacent angles..

B Q70º 20º

Another illustration:Another illustration:

mm R + m R + m P = 180.P = 180. R and R and P P areare supplementary angles supplementary angles

and and non - adjacent anglesnon - adjacent angles..

R P30º

150º


RPD and RPD and QPD QPD are are supplementary angles supplementary angles and and Adjacent Adjacent

anglesangles..

R P Q

D

35º145º


They are non- common sides & opposite They are non- common sides & opposite rays. rays.

RPD and RPD and QPD are LINEAR PAIR of QPD are LINEAR PAIR of angles.angles.

R P Q

D

35º145º

What can you say about ray PR & What can you say about ray PR & ray PQ of ray PQ of RPD & RPD & QPD?QPD?

Definition of LINEAR PAIRDefinition of LINEAR PAIR

Are TWO adjacent angles and whose Are TWO adjacent angles and whose non common sides are opposite non common sides are opposite rays.rays.

LINEAR PAIR POSTULATELINEAR PAIR POSTULATEStates that “ Linear pair of angles are States that “ Linear pair of angles are

supplementary”supplementary”

In the figure:In the figure:

RPD and RPD and QPD are QPD are LINEAR PAIR LINEAR PAIR of angles and of angles and supplementarysupplementary..

R P Q

D

35º145º

In the figure, name & identify In the figure, name & identify linear pair of angles.linear pair of angles.

APC and APC and BPCBPC, , APD and APD and APC APC APD and APD and DPBDPB, , DPD and DPD and BPC BPC are LINEAR PAIR of angles.are LINEAR PAIR of angles.

A

BC

D

P

REMEMBER THIS…..REMEMBER THIS…..

LINEAR PAIR of angles LINEAR PAIR of angles are are adjacent and adjacent and supplementarysupplementary..

In the figure, we can write an In the figure, we can write an equation. Like,equation. Like,

mmAPC +mAPC +mBPC = 180BPC = 180

mmAPD + mAPD + mAPC = 180APC = 180

mmAPD + mAPD + mDPB = 180DPB = 180

mmDPD + mDPD + mBPC = 180BPC = 180

A

BC

D

P

In the figure, if In the figure, if mmAPD = 120. APD = 120. . What . What is the measure of the other angles?is the measure of the other angles?

mmAPC +mAPC +mBPC = 180BPC = 180

mmAPD APD + m+ mAPC = 180APC = 180

mmAPD APD + m+ mDPB = 180DPB = 180

mmDPD + mDPD + mBPC = 180BPC = 180

A

BC

D

P

In the figure, if In the figure, if mmAPD = 120. APD = 120. . What . What is the measure of the other angles?is the measure of the other angles?

mmAPD APD + m+ mAPC = 180 APC = 180 (linear pair postulate)(linear pair postulate)

120 + m120 + mAPC = 180 APC = 180 ( by substitution)( by substitution)

mmAPC = 60APC = 60( by subtraction)( by subtraction)

A

BC

D

P60°

120°

120°

60°

In the given figure, what are In the given figure, what are non- non- adjacent anglesadjacent angles??

APD APD and and BPC BPC

APC and APC and BPDBPDThese non-adjacent angles are also called These non-adjacent angles are also called vertical angles.vertical angles.

A

BC

D

P60°

120°

120°

60°

Vertical AnglesVertical Angles

In the figure, In the figure, APC and APC and BPD, BPD, APD APD and and BPC are vertical angles.BPC are vertical angles.

A

BC

D

P

Vertical AnglesVertical AnglesARE TWO NON ARE TWO NON ADJACENT ANGLES ADJACENT ANGLES formed by two formed by two intersecting lines.intersecting lines.APC and APC and BPD, BPD,

APD and APD and BPC are BPC are NON NON ADJACENT anglesADJACENT angles..

Line AB Line AB and and line CD line CD are two intersecting are two intersecting lineslines

A D

C

P

B

What can you say about the What can you say about the measures of the measures of the vertical anglesvertical angles??

mmAPD APD = m= mBPC BPC

mmAPC = mAPC = mBPDBPD

A

BC

D

P60°

120°

120°

60°

APD APD and and BPC BPC

APC and APC and BPDBPDThese non-adjacent angles are also called These non-adjacent angles are also called vertical angles.vertical angles.

Vertical Angles TheoremVertical Angles Theorem

Vertical angles are Vertical angles are congruent.congruent.

Fixing skills

• In the given figure,

APB and CPD are right angles.

Name all pairs of:1. Complementar

y angles.

1

5

3

2

4

6

7 8C

B

P

A

D

ANSWERS:

3 AND 3 AND 445 AND 5 AND 66

Fixing skills



Name all pairs of:

2. Supplementary angles.

1

5

3

2

4

6

7 8C

B

P

A

D

ANSWERS:


Fixing skills



Name all pairs of:3. Vertically

opposite angles.

1

5

3

2

4

6

7 8C

B

P

A

D

ANSWERS:


CPD and CPD and BPABPAAPC and APC and DPBDPB

Fixing skills



Name all pairs of:

4. Linear pair of angles.

1

5

3

2

4

6

7 8C

B

P

A

D

ANSWERS:


Fixing skills



Name all pairs of:

5.Adjacentangles.

1

5

3

2

4

6

7 8C

B

P

A

D

ANSWERS: 1 AND 1 AND 223 AND 3 AND 445 AND 5 AND 667 AND 7 AND 88

STUDENT ACTIVITY

Define the following pairs of angles:

•Adjacent angles•Linear pair of angles•Vertical angles

State the following:

•Linear pair postulate•“Linear pair of angles are

supplementary”•Vertical angles Theorem

•“Vertical angles are congruent”

QUIZ

Get ¼ sheet of paper

State whether each of thefollowing is TRUE or FALSE.1. TWO ADJACENT RIGHT ANGLES

ARE SUPPLEMENTARY.2. ALL SUPPLEMENTARY ANGLES

ARE ADJACENT.3. SOME SUPPLEMENTARY ANGLES

ARE LINEAR PAIR.

State whether each of thefollowing is TRUE or FALSE.

4. TWO VERTICAL ANGLES ARE ALWAYS CONGRUENT.

5. ALL RIGHT ANGLES ARE CONGRUENT.

ASSIGNMENT

•A. Copy & answer the following. Show your solution .

• ¼ sheet of paper

Find the value of x.

4x – 20 2x + 10

1.


x + 35 2x - 5

2.

ASSIGNMENT

• B. DEFINE THE FOLLOWING:• - PERPENDICULAR LINES• - PERPENDICULAR BISECTOR• - EXTERIOR ANGLE OF A TRIANGLE• NOTE:• Write your answer in your notebook

B. Solve the following

•The supplement of a certain angle is four times larger than its complement. What is the measure of the angle?


5 x x

3.

4x


2x + 10

4..

3x-5

B. Solve the following

•Two complementary angles are on the ratio 1 : 4. What is the measure of the larger angle?

Solution • Let x = the measure of the smaller

angle• 4x = the measure of the larger

angle

• X + 4x = 90• 5x = 90• X = 18•Therefore, the measure of

the larger angle is 4 ( 18 ) or 72.

SOLVE AND CHECK• The measure of

one of two complementary angles is 15 less than twice the measure of the other. Find the measure of each angle.

Solution:Let x = measure

of the first angle

2x – 15 = measure of the second

angle.

Solution:

X + 2x – 15 = 90 (def. of complementary angles)

3x = 90 + 15 3x = 105 x = 35 ( measure of the 1st

) 2x -15 = 55 (measure of the 2nd

)

• The measure of one of two complementary angles is 15 less than twice the measure of the other. Find the measure of each angle.

Check: 55 is 15 less than twice 35. the

sum of 35 and 55 is 90.

OPENING ACTIVITY GEOMETRY JEOPARDY GEOMETRY JEOPARDY.

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