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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
Consider the following:Consider the following:
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°.
Consider the following:Consider the following:
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..
Consider this figureConsider this figure
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
Consider this figureConsider this figure
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
Consider the following:Consider the following:
RPD and RPD and QPD QPD are are supplementary angles supplementary angles and and Adjacent Adjacent
anglesangles..
R P Q
D
35º145º
Consider the following:Consider the following:
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
• In the given figure,
APB and CPD are right angles.
Name all pairs of:
2. Supplementary angles.
1
5
3
2
4
6
7 8C
B
P
A
D
ANSWERS:
1 AND 1 AND 227 AND 7 AND 88
Fixing skills
• In the given figure,
APB and CPD are right angles.
Name all pairs of:3. Vertically
opposite angles.
1
5
3
2
4
6
7 8C
B
P
A
D
ANSWERS:
3 AND 3 AND 665 AND 5 AND 44
CPD and CPD and BPABPAAPC and APC and DPBDPB
Fixing skills
• In the given figure,
APB and CPD are right angles.
Name all pairs of:
4. Linear pair of angles.
1
5
3
2
4
6
7 8C
B
P
A
D
ANSWERS:
1 AND 1 AND 227 AND 7 AND 88
Fixing skills
• In the given figure,
APB and CPD are right angles.
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.
Find the value of x.
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?
Find the value of x.
5 x x
3.
4x
Find the value of x.
2x + 10
4..
3x-5
Find the value of x.
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.
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