Theorem of 9th
Congruent Triangles
SLOS 9.1 Congruent Triangles
9.1.1 apply the following theorems to solve related problems.
 In any correspondence of two triangles, if one side and any two angles of one
Triangle is congruent to the corresponding side and angles of the other, the two triangles
are congruent.
 if two angles of a triangle are congruent then the sides opposite to them arealso
congruent.
 in a correspondence of two triangles, if three sides of one triangle arecongruent to the
corresponding three sides of the other, the two triangles arecongruent.
 if in the correspondence of two right-angled triangles, the hypotenuse and oneside of
one are congruent to the hypotenuse and the corresponding side of theother, then the
triangles are congruent
•
•
•
•
•
•
•
•
•
•
•
•
Definitions
Triangle: three-sided figure
Polygon: a closed figure in a plane that is made up of segments
Sides: the segments which make up a polygon
Vertices: The endpoints of the sides
Acute Triangle: All angles measure less than 90
Obtuse Triangle:
One angle measures more than 90
Right Triangle: One angle measures 90
Equiangular Triangle: All angles are congruent
Equilateral Triangle: All sides are congruent
Scalene Triangle: No two sides are congruent.
Isosceles Triangle: at least two sides are congruent
Page 1 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
The Concept of Congruence
Congruence - Congruent
You walk into your favorite mall and see dozens
of copies of your favorite CD on sale. All of the
CDs are exactly the same size and shape.
In fact, you can probably think of many objects
that are mass produced to be exactly the same size
and shape.
Objects that are exactly the same size and shape are said to be
congruent.
Congruent objects are duplicates of one another.
If two mathematical figures are congruent and you cut one figure out with a pair of scissors, it will fit
perfectly on top of the other figure.
Mathematicians use the word congruent to describe geometrical
figures.
-- If two quadrilaterals (4 sided) are the same size and shape,
they are congruent.
-- If two pentagons (5 sided) are the same size and shape,
they are congruent.
-- If two polygons (any number of sides) are the same size and
shape,
they are congruent.
-- If two line segments are the same length (they already are the
same
shape), they are congruent.
The mathematical symbol used to denote
congruent is .
The symbol is made up of two parts:
which means the same shape (similar) and
which means the same size (equal).
Congruent
Symbol
Page 2 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
When you are looking at congruent figures, be sure to find the sides
and the angles that "match up" (are in the same places) in each
figure. Sides and angles that "match up" are called corresponding
sides and corresponding angles.
In congruent figures, these corresponding parts are also congruent.
The corresponding sides will be equal in measure (length) and that
the corresponding angles will be equal in degrees.
Theorems for Congruent Triangles
Latest news bulletin:
The most popular congruent figures are triangles!
In many geometrical proofs, it may be necessary to prove that
two triangles are congruent to each other. The task may simply
be to prove the triangles congruent, or it may be to use these
congruent triangles to gain additional information.
When triangles are congruent and one triangle is placed on top of the other,
the sides and angles that coincide (are in the same positions) are called corresponding parts.
Example:
When two triangles are congruent, there are
6 facts that are true about the triangles:


the triangles have 3 sets of congruent
(of equal length) sides and
the triangles have 3 sets of congruent
NOTE: The corresponding
congruent sides are marked
with small straight line
segments called hash marks.
The corresponding congruent
angles are marked with arcs.
Page 3 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
(of equal measure) angles.
The 6 facts for our congruent triangles example:
Note:
The order of the letters in the names of the triangles should display the corresponding
relationships. By doing so, even without a picture, you would know that <A would be congruent to
<D, and
would be congruent to
, because they are in the same position in each triangle
name.
Wow! Six facts for every set of congruent triangles!
Fortunately, when we need to PROVE (or show) that triangles are congruent, we do NOT need to
show all six facts are true. There are certain combinations of the facts that are sufficient to prove that
triangles are congruent. These combinations of facts guarantee that if a triangle can be drawn with
this information, it will take on only one shape. Only one unique triangle can be created, thus
guaranteeing that triangles created with this method are congruent.
Methods for Proving (Showing) Triangles to be Congruent
SSS
If three sides of one triangle are congruent to three sides of
another triangle, the triangles are congruent.
(For this method, the sum of the lengths of any two sides must be greater than
the length of the third side, to guarantee a triangle exists.)
SAS
If two sides and the included angle of one triangle are
congruent to the corresponding parts of another triangle,
the triangles are congruent. (The included angle is the angle formed
by the sides being used.)
Page 4 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
ASA
If two angles and the included side of one triangle are
congruent to the corresponding parts of another triangle,
the triangles are congruent. (The included side is the side between
the angles being used. It is the side where the rays of the angles would
overlap.)
AAS
If two angles and the non-included side of one triangle are
congruent to the corresponding parts of another triangle,
the triangles are congruent. (The non-included side can be either of
the two sides that are not between the two angles being used.)
HL
Right
Triangles
Only
If the hypotenuse and leg of one right triangle are
congruent to the corresponding parts of another right
triangle, the right triangles are congruent. (Either leg of the right
triangle may be used as long as the corresponding legs are used.)
BE CAREFUL!!!
Only the combinations
listed above will give
congruent triangles.
So, why do other combinations not work?
Methods that DO
AAA
NOT Prove Triangles to be Congruent
AAAworks fine to show that triangles are the same
SHAPE (similar), but does NOT work to also show
they are the same size, thus congruent!
You can easily draw 2 equilateral
triangles that are the same shape but
are not congruent (the same size).
Consider the example at the right.
Page 5 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
SSA
or
ASS
SSA (or ASS)is humorously referred to as the
"Donkey Theorem".
This is NOT a universal method to prove triangles
congruent because it cannot guarantee that one unique
triangle will be drawn!!
The SSA (or ASS) combination affords the possibility of creating
zero, one, or two triangles. Consider this diagram of triangle DEF. If
for the second side, EF is equal to EG(the minimum distance needed to
create a triangle), only one triangle can be drawn. However, if EF is
greater than EG, two triangles can be drawn as shown by the dotted
segment. Should EF be less than the minimum length needed to
create a triangle, EG, no triangle can be drawn.
The possible "swing" of side
can create two different triangles which causes our
problem with this method. The first triangle, below, and the last triangle both show SSA,
but they are not congruent triangles.
The combination of SSA (or ASS) creates a unique triangle ONLY when working in a right
triangle with the hypotenuse and a leg. This application is given the name HL(HypotenuseLeg) for Right Triangles to avoid confusion. You should not list SSA (or ASS) as a reason
when writing a proof.
Once you prove your triangles are congruent, the "left-over" pieces that
were not used in your method of proof, are also congruent.
Remember, congruent triangles have 6 sets of congruent pieces. We
now have a "follow-up" theorem to be used AFTER the triangles are
known to be congruent:
Theorem: (CPCTC) Corresponding parts
of congruent triangles are congruent.
Page 6 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Tips for Working with Congruent Triangles in Proofs
Two triangles are congruent if all pairs of corresponding sides are congruent, and all pairs of
corresponding angles are congruent. Fortunately, we do not need to show all six of these congruent
parts each time we want to show triangles congruent. There are 5 combination methods that allow us
to show triangles to be congruent.
Remember to look for ONLY these combinations
for congruent triangles:
SAS, ASA, SSS, AAS, and HL(right triangle)
But how do we decide which method we should be using?
Let's look at some examples and tips:
Example 1:
Here is an example problem, using one of the methods mentioned above.
Prove:
:
Which congruent triangle method do you think
is used in this example?
Did you notice that the congruent triangle parts
that were given to us were marked up in the
diagram? This technique is very helpful when
trying to decide which method of congruent
triangles to use.
TIP: Mark any given information on your diagram.
Mark diagram
Page 7 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Example 2:
In this example problem, examine the given information, mark the given information on the diagram
as in the first tip, and decide if congruent triangles will help you solve this problem.
Prove:
:
This problem does not ask you to prove the triangles are congruent. This, however, does not mean that you
should not "look" for congruent triangles in this problem. Remember that once two triangles are congruent,
their "left-over" corresponding pieces are also congruent. If you can prove these two triangles are
congruent, you will be able to prove that the segments you need are also congruent since they will be "leftover" corresponding pieces.
Which of the congruent triangle methods
do you think is used in this example?
For the triangles in this second example, three sets of corresponding parts were used to prove the
triangles congruent. Can you name the other 3 sets of corresponding parts?
CLICK HERE to see the answer.
Corresponding
Parts
TIP: Look to see if the pieces you need are "parts"
of the triangles that can be proven congruent.
Example 3:
In this example problem, examine the given information, decide what else you need to know, and
then decide the proper method to be used to prove the triangles congruent.
Prove:
Page 8 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
There seems to be missing information in this problem. There are only two pieces of congruent information
given. This problem expects you to "find" the additional information you will need to show that the
triangles are congruent. What else do you notice is true in this picture?
Which of the congruent triangle methods
do you think is used in this example?
Examine
Diagram
TIP: If not given all needed pieces to prove the
triangles congruent, look to see what else you might
know about the diagram.
Example 4:
In this example problem, examine the given information carefully, mark up the diagram and then
decide upon the proper method to be used to prove the triangles congruent.
When you marked up the diagram, did you mark the information gained from the definition of the angle
bisector? While this problem only gives you two of the three sets of congruent pieces needed to prove the
triangles congruent, it also gives you a "hint" as to how to obtain the third needed set. The "hint" in this
problem is in the form of a definition - the angle bisector.
Which of the congruent triangle methods do you
think is used in this example?
Use Definitions
TIP: Know your definitions! If the given
information contains definitions, consider these as
"hints" to the solution and be sure to use them.
This particular example can be solved in more than one way.
Page 9 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Even though the given information gives congruent information about <B and <D, this information is not needed
to prove the triangles congruent. The two triangles in this problem "share" a side (called a common side). This
"sharing" automatically gives you another set of congruent pieces.
More than one
solution
Common Parts
TIP: Stay open-minded. There may be more than
one way to solve a problem.
TIP: Look to see if your triangles "share" parts.
These common parts are automatically one set of
congruent parts.
In summary, when working with congruent triangles, remember to:
1. Mark any given information on your diagram.
2. Look to see if the pieces you need are "parts" of the triangles that can be proven congruent.
3. If not given all needed pieces to prove the triangles congruent, look to see what else you might
know about the diagram.
4. Know your definitions! If the given information contains definitions, consider these as "hints" to
the solution and be sure to use them.
5. Stay open-minded. There may be more than one way to solve a problem.
6. Look to see if your triangles "share" parts. These common parts are automatically one set of
congruent parts.
Remember that proving triangles congruent is like solving a puzzle. Look carefully at the
"puzzle" and use all of your geometrical strategies to arrive at an answer.
Page 10 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Numerical Practice with Congruence
Answer the following questions dealing with congruent figures.
1.
2.
with
and
AB = 10 inches, which side of
same measure?
. If
has the
Choose:
Triangle BAT is congruent to triangle FOG with
Choose:
BA = FO, AT = OGand BT = FG.
If the sum of the measures of
and
is 110 degrees, what
70
is the degree measure of
?
80
90
110
3.
The two legs of a right triangle measure 5 and 12. The leg and Choose:
hypotenuse of another right triangle measure 12 and 13
respectively. These two triangles are congruent.
True
False
Page 11 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Choose:
4.
Find AB
(explain your answer)
15
30
45
60
5.
Choose:
and the perimeter of
is 30
inches. If the sum of two sides of
is 23 inches, what
7 inches
is the length of the third side of
?
10 inches
23 inches
30 inches
6.
Triangle ABC is congruent to triangle DEF
with
.
If
AB = 2x + 10, and DE = 4x - 20,
find the value of x.
Choose:
5
10
15
60
Page 12 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
7.
8.
Which of the following cannot be used in proving triangles
congruent?
a. SSS SSS
b. AAA AAA
c. SAS SAS
d. AAS AAS
Triangle ABC is congruent to triangle A'B'C'.
If angle C is represented by 2x - 10
and angle C' is represented by 3x - 40,
find the measure of angle C.
Choose:
a
b
c
d
Choose:
15
30
50
90
9.
Triangle DEF is congruent to triangle D'E'F'.
If EF is represented by 6x + 4
and E'F' is represented by 2x + 20,
find the value of x.
Choose:
3
4
5
6
Page 13 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Two triangles are congruent if they have 2 angles and the
included side of one triangle congruent to the corresponding
parts of the other triangle.
10.
Choose:
True
False
11.
If three sides of one triangle are congruent
Choose:
respectively to the three sides of another triangle, then
the 2 triangles are _________ congruent.
Always
Sometimes
Never
13.
Polygon ABCD is congruent to polygon
A'B'C'D'.
If angle A corresponds to angle A',
the value of angle A' is 5x - 65 and
the value of angle A is 2x - 5,
find angle A.
Choose:
10
20
30
35
Page 14 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
(NP ==> Not Possible to prove congruent triangles with the information given.)
1.
SAS
ASA
SSS
AAS
HL
NP
ASA
SSS
AAS
HL
NP
ASA
SSS
AAS
HL
NP
2.
SAS
3.
SAS
Page 15 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
4.
SAS
ASA
SSS
AAS
HL
NP
ASA
SSS
AAS
HL
NP
ASA
SSS
AAS
HL
NP
5.
SAS
6.
SAS
Practice with
Proofs Involving Congruent Triangles
Page 16 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Write a formal proof for the following problems.
(Please Note: The proofs presented as answers will show only ONE possible solution to each
problem. Keep in mind that there is often more than one way to solve a problem and more than one
manner of presentation. Two-column proofs will be presented here.)
1.
2.
3.
Page 17 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
4.
5.
6.
Page 18 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
7.
8.
9.
Page 19 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
10.
Given:
Which method can NOT be used to prove:
Choose:
SSS
SAS
AAS
HL
Parallelograms and Triangles
10.1.1 apply the following theorems to solve related problems
 in a parallelogram:
i) the opposite sides are congruent,
ii) the opposite angles are congruent,
iii) the diagonals bisect each other.
 if two opposite sides of a quadrilateral are congruent and parallel, it is a
parallelogram.
 the line segment, joining the midpoints of two sides of a triangle, is
parallel to the third side and is equal to one half of its length.
 the medians of a triangle are concurrent and their point of concurrency is
the point of trisection of each median.
 if three or more parallel lines make congruent intercepts on a transversal
they also intercept congruent segments on any other transvers
Page 20 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 21 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 22 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 23 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 24 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 25 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
PARALLELOGRAMS WORKSHEET
Complete each statement
1. In a parallelogram, opposite sides are _________________ and ____________
2. In a parallelogram, consecutive angles are _______________
3. In a parallelogram, diagonals __________ each other, which means they split each other in _________
Complete each statement, using Parllelogram DCBA
4. If AD = 20, then BC = ____
5. If AB = 13, then DC = ______
6. If DB = 22, then DE = ______
7. If AE = 18, then AC = ______
8. If m<ADC = 115o, then m<ABC = ______
9. If m<DAB = 75o, m<ADC = ______
then m<4 = ______
11. If m<AED= 72o, m<DEC = ______
m<1 = 30o,
10. If
12. If m<ADC= 130o, and m<1= 35o, m<2= _____
13. If AC = 30 and AE = 3x + 3, then x
14. If DC = 6x + y, BC = 3x + 2y, AB = 25, and AD =
= _____
14, then x = _____ and y = _____
Find the missing measurements of Parallelogram ADCB.
AB = 10 BC = 22 AE =12 BE = 13
m<ABE = 47o
m<EBC = 23o
m<ECD = 72o CD = ______ DA = _____
AC = _____
DB = _____
CE = ______
DE = _____
m< ABC = _____ m<BCE = _____ m< BCD = _____ m<ADC = _____ m<BAD = ____ m<CDE = _____
_____ m<DAE = _____ m<EAB = _____ m<AEB = ____
m<BEC = _____
m<CED = _____
m<EDA =
m<DEA = _____
Page 26 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Parallelogram Worksheet
Make a checkmark to indicate which parallelograms below (rectangle, rhombus, square, etc…) have each given property. If
all three types (rectangle, rhombus, square) have a given property, then put a check mark under “All Parallelograms.”
Property
Rectangle
Rhombus
Square
All
Parallelograms
The Sides
Opposite sides are parallel
Opposite sides are congruent
Al sides are congruent
The Angles
Sum of the angles is 360°
Opposite angles are congruent
All four angles are right angles
The Diagonals
Diagonals bisect each other
Diagonals are Congruent
Diagonals are perpendicular
Diagonals bisect opposite angles
Page 27 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 28 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 29 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 30 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 31 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Activity I
In the following figure, D and E are the midpoints of side AB and AC.
Measure the following quantities:
(1)m ABC , m ADE
(2) |DE| and |BC|.
A
E
D
C
B
What can you say about DE and BC? Why?
So DE// BC (Why?) and the length of DE is half of that of BC.
Mid-Segment of a Triangle
Definition:
The mid-segment of a triangle (also called a midline) is a
segment joining the midpoints of two sides of a triangle.
Properties:
1.
The mid-segment of a triangle joins the midpoints of two sides of a triangle
such that it is parallel to the third side of the triangle.
2. The mid-segment of a triangle joins the midpoints of two sides of a triangle
such that its length is half the length of the third side of the triangle.
Page 32 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Examples:
1. Given DE is the length of the midsegment. Find AB.
2. Given DE, DF, and FE are the
lengths of mid-segments. Find the
perimeter of triangle ABC.
Solution:
The mid-segment is half of the third side.
7 is half of 14.
AB = 14.
Solution:
The mid-segment is half of the third side.
6 is half of 12 so AC = 12
7 is half of 14 so CB = 14
8 is half of 16 so AB = 16
The perimeter of the large triangle ABC is:
12 + 14 + 16 = 42.
Page 33 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Example 1
Suppose that in the following figure,points D and E are the midpoints of AB and AC respectively,
mADE  61, mACB  58 and AE = 2.51cm.
A
Find BAC , BDE and | AC |.
D
B
E
C
Page 34 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 35 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 36 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 37 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 38 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 39 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 40 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Line Bisectors and Angle Bisectors
 any point on the right bisector of a line segment is equidistant from its end
points.
 any point equidistant form the end points of a line segment is on the
rightbisector of it.
 The right bisectors of the sides of a triangle are concurrent.
 any point on the bisector of an angle is equidistant from its arms
 any point inside an angle, equidistant from its arms, is on the bisector of itthe
bisector of the angles of a triangle are concurrent.
Page 41 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Theorem
Any point on the right bisector of a line segment is equidistant from its end points.
Converse of theorem
•Any point equidistant from the points of a line segment is on the right bisector of it.
̅̅̅̅ = ____________
1. 𝑚𝐴𝑃
̅̅̅̅ = ___________
2. 𝑚𝐵𝑄
̅̅̅̅ =_____________
3. 𝑚𝐶𝑃
Page 42 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Theorem
Any point on the bisector of an angle is equidistant from its arms
Converse of theorem
Any point inside an angle, equidistant from its arms, is on the bisector of it
Page 43 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Perpendicular Bisectors of a Triangle
Page 44 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 45 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 46 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 47 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 48 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 49 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
(pre-made sample)
Date ___________________
Name _____________________________
1.
and
are angle bisectors. If m
find the measure of BPC.
(Key 1 - Answer ID # 0485345)
A = 76 ,
2. If k n, ABC is an isosceles, m 1 = 4x and m
2 = (90 -x) , find the measure of m BAC.
3. Which of the following statements is NOT correct? 4. In
and
bisect ADE and CED
respectively. If m CED = 113 and m
71 , find m ABF.
(III)
is the bisector of
BAC.
I and III only
II only
I only
III only
I, II, and III
6.
DFE =
is the perpendicular bisector of
and D lies on
. Which statement(s) must
be true?
(I) ABD
ACD
(II) ABC is equilateral.
In an isosceles triangle, the altitude to the
base bisects the base and the vertex angle.
The shortest distance from a vertex to the
opposite side is the altitude to that side.
The circumcenter of a triangle is always
inside the triangle.
An equilateral triangle is also an equiangular
triangle.
5.
ABC,
In
ABC,
B = 66 , and m
and
bisects BAC, m
C = 36 . Find m DAE.
Page 50 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
(pre-made sample)
Date ___________________
Name _____________________________
1. If O is the circumcenter of
, find m AOB.
ABC and m
(Key 1 - Answer ID # 0430138)
C = 39 2. The perpendicular bisectors of
point G. If
= 12,
ABC meet at
= 6, and
= 3, find
.
5.
Medians
,
intersects
, and
at H, and
of
ABC meet at G,
= 18. Find
.
6. In the figure, P is the in center of ABC, the radius
of the inscribed circle is 4 cm, and the perimeter of
ABC is 41 cm. What is the area of ABC?
Page 51 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
(pre-made sample)
Date ___________________
Name _____________________________
1.
In
PQR,
S lies on
, and
bisects
QPR and
. Which statement(s) must be true?
is a perpendicular bisector of
is perpendicular to
of
(Key 1 - Answer ID # 0292230)
2. In
ABC, k is the perpendicular bisector of side
and
=
. Which of the following
statements is incorrect?
.
but not a bisector
.
is a bisector of
perpendicular to
but not
.
is neither a bisector of
perpendicular to
Point D is the circumcenter of ABC.
ABC is a right triangle.
DAB
DBA
Point D is the incenter of ABC.
nor
.
4. X is a point inside triangle ABC. If X is equidistant
from
and
bisector of
, then X must lie on the ______.
C
perpendicular bisector of
median of C
altitude to side
Page 52 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 53 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 54 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 55 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 56 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 57 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 58 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Sides and Angles of a Triangle
 if two sides of a triangle are unequal in length, the longer side has an angle of
greater measure opposite to it.
 if two angles of a triangle are unequal in measure, the side opposite to the
greater angle is longer than the side opposite to the smaller angle.
 the sum of the lengths of any two sides of a triangle is greater than the length of
the third side.
 from a point, out-side a line, the perpendicular is the shortest distance from the
point to the line.
Page 59 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Triangle Inequalities
Theorem 1:
The sum of the lengths of any two sides of a triangle
must be greater than the third side.
If these inequalities are
NOT true, you do not
have a triangle!
Example
Suppose we know the lengths of two sides of a
triangle, and we want to find the "possible"
lengths of the third side.
According to our theorem, the following 3 statements must be true:
5 + x> 9
5 + 9 >x
x+9>5
So,x> 4
So,14 >x
So, x> -4
(no real information is gained here since the
lengths of the sides must be positive.)
Putting these statements together, we get that xmust begreater than 4,butless than 14. So any
number in the range 4 <x< 14 can represent the length of the missing side of our triangle.
Page 60 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Theorem 2:
also ...
Theorem 3:
In a triangle, the longest side is across from the
largest angle.
In a triangle, the largest angle is across from the
longest side.
Since 7 is the longest side in the
triangle, <C, across from it, is the
largest angle.
Since 100° is the largest angle in this
triangle,
, across from it, is the longest
side.
These theorems can be modified to apply to a discussion of only two angles within the triangle:
Theorem: In a triangle, the longer side is across from the larger angle.
Theorem: In a triangle, the larger angle is across from the longer side.
Example
Suppose we want to know which side of this
triangle is the longest.
Before we can utilize our theorem, we need to
know the size of <B. We know that the 3 angles
of the triangle add up to 180°.
We have now found that <B measures 60°.
According to our theorem, the longest side will be
across from the largest angle.
80 + 40 + x = 180
120 + x = 180
x = 60
Now that we know the measures of all 3
angles, we can tell that <A is the
largest. This means the side across from
<A,side
, is the longest side.
Page 61 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Answer each of the following questions related to triangle inequalities.
1.
Which of the following could represent the lengths of the sides of a triangle ?
Choose one:
1, 2, 3
6, 8, 15
5, 7, 9
2.
Two sides of an isosceles triangle measure 3 and 7. Which of the following could
be the measure of the third side ?
Choose one:
9
7
3
3.
In triangle ABC,m<A = 30° and m<B = 50°. Which is the longest side of the triangle
?
Choose one:
Page 62 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
4.
In triangle DEF, an exterior angle at D measures 170°, and m<E = 80°. Which is the
longest side of the triangle ?
Choose one:
5.
In triangle ABC, m<C = 55°, and m<C > m<B. Which is the longest side of the
triangle ?
Choose one:
Page 63 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
IF two sides of a triangle are unequal, the angle opposite to thelonger
side is larger (or greater).
CONVERSE
If
two angles of a triangle are
unequal in measure, the side
opposite to the greater angle
is longer than the side opposite to the smaller angle
ACTIVITY
Page 64 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
1.
2.
3.
Page 65 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 66 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 67 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 68 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Application of Ratio and Proportion in Geometrical Theorems
 a line parallel to one side of a triangle, intersecting the other two sides, divides
them proportionally.
 if a line segment intersects the two sides of a triangle in the same ratio then it is
parallel to the third side.
 the internal bisector of an angle of a triangle divides the side opposite to it’s in
the ratio of the length of the sides containing the angle.
 if two triangles are similar, the measures of their corresponding sides are
proportional.
Page 69 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Similar Triangles
Objects, such as these two cats, that have the same shape, but
do not necessarily have the same size, are said to be "similar".
The cat on the right is an
enlargement of the cat on the left.
They are exactly the same shape, but
they are NOT the same size.
These cats are similar figures.
The mathematical symbol used to denote
similar is .
Similar
Symbol
Do you remember this symbol as "part" of the
symbol for congruent??
Definition: In mathematics, polygons are similar if their corresponding
(matching) angles are congruent (equal in measure) and the ratio of their
corresponding sides are in proportion.
(This definition allows for congruent figures to also be "similar",
where the ratio of the corresponding sides is 1:1.)
Page 70 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Facts about similar triangles:
The ratio of the corresponding
sides is called the ratio of
similitude or scale factor.
Strategies for Dealing with Similar Triangles
Triangles are similar if their corresponding (matching) angles are congruent (equal
in measure) and the ratio of their corresponding sides are in proportion.
Page 71 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
There are many different types of
problems that involve similar
triangles. And, fortunately, there
are several different ways to arrive
at an answer.
Keep an open mind!
Remember that there
may be more than one
way to arrive at an
answer!
Let's look at some strategies for arriving at answers!
Style 1:
The similar triangles are two separate triangles:
Find x:
Create a proportion
matching the
corresponding sides.
Two possible answers:
Small
triangle on
top:
Large triangle
on top:
x = 20
x = 20
HINT: These two triangles are sitting such that their corresponding parts are in the
same position in each triangle. If the triangles are not sitting in this manner, you can
match the corresponding sides by looking across from the angles which are equal in each
triangle.
Style 2:
The similar triangles overlap:
Page 72 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Many problems involving similar triangles
have one triangle ON TOP OF (overlapping)
another triangle.
Since
is marked to be parallel to
, we
know that we have <BDE congruent to
<DAC (by corresponding angles). <B is
shared by both triangles, so the two triangles
are similar by AA.
There are two ways to attack this type of problem.
Use FULL sides of the two triangles
when dealing with the problem. Do not
use DA or EC since they are not sides of
triangles.
EASIEST METHOD TO USE
Use a theorem relating to parallel lines, which
says that If a line is parallel to one side of a
triangle, and intersects the other two sides, the
line divides these two sides proportionally.
EASY TO FORGET!!
Let's try some problems with this type of diagram:
Find BE:
Read carefully to see WHAT you are
supposed to find. This problem asks you
to find BE.
Here are two solutions letting BE = x.
Use FULL sides of the Use the theorem related
triangles, cross
to parallel lines, cross
multiply and solve.
multiply and solve.
4x + 36 = 12x
36 = 8x
4.5 = x
36 = 8x
4.5 = x
This problem asks you to find EC.
Here are two solutions letting EC = x:
Find EC:
Use FULL sides of the
triangles, cross multiply
and solve.
Use the theorem
related to parallel
lines, cross multiply
Page 73 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
and solve.
32 + 4x = 80
4x = 48
x = 12
Find x:
4x = 48
x = 12
CAREFUL!!!
This problem MUST use the full sides of
triangles as a solution. The parallel
theorem does not work here. The problem
asks you to find x where x is a FULL side.
Here is the solution:
x=5
Page 74 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Answer the following questions dealing with similar figures.
2.
The sides of a triangle are 5, 6 and 10. Find the length of the Choose:
longest side of a similar triangle whose shortest side is 15.
10
15
18
30
4.
Given: In the diagram,
is parallel to
, BD = 4,
DA = 6 and EC = 8. Find
BC to the nearest tenth.
Choose:
4.3
5.3
8.3
13.3
Page 75 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
5.
Find BC.
Choose:
4
4.5
13.5
17
6.
Two ladders are leaned against a Choose:
wall such that they make the
same angle with the ground. The
4.5'
10' ladder reaches 8' up the wall.
6.4'
How much further up the wall
14.4'
does the 18' ladder reach?
22.4'
8.
Two triangles are similar. The sides of the first triangle are 7, Choose:
9, and 11. The smallest side of the second triangle is 21. Find
the perimeter of the second triangle.
27
33
63
81
Page 76 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
9. Two triangular roofs are similar.
The
corresponding sides of these roofs are
16 feet and 24 feet. If the altitude of
the smaller roof is 6 feet, find the
corresponding altitude of the larger
roof.
Choose:
6
9
36
81
Page 77 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Answer the following questions dealing with similar triangles.
2.
As marked, by which method would it be
possible to prove these triangles similar (if
possible)?
Choose:
AA
SSS
SAS
not similar
3. As marked, by which method would it be
possible to prove these triangles similar
(if possible)?
Choose:
AA
SSS
SAS
not similar
Page 78 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
4.
As marked, by which method would it be possible to
prove these triangles similar (if possible)?
Choose:
AA
SSS
SAS
not similar
6.
Proof
Page 79 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 80 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 81 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 82 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 83 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 84 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 85 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 86 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com
Page 87 of 87
MADE BY ARSLAN SHAIKH AGAKHAN SCHOOL HYDERABAD
Sheikhs7@hotmail.com