Home MATHEMATICS TOPIC 5: CONGRUENCE ~ MATHEMATICS FORM 2

TOPIC 5: CONGRUENCE ~ MATHEMATICS FORM 2

4182
12
TOPIC 11: STATISTICS ~ MATHEMATICS FORM 2
TOPIC 5: CONGRUENCE ~ MATHEMATICS FORM 2

TOPIC 5: CONGRUENCE ~ MATHEMATICS FORM 2

Congruence

I’m sure you have seen some of the figure which in one way or another one of the shape can become another using turns, flip or slide.

These shapes are said to be Congruent.

Study this notes carefully to know different ways that can help you to recognize congruent figures.

In geometry two figures or objects are congruent if they have the same shape and size, or if one has the shape and size as the mirror image of the other.
If one shape can become another using turns (rotation), flip (reflection), and/ or slide (translation), then the shapes are Congruent.
After any of these transformations the shape must still have the same size, perimeters, angles, areas and line lengths.
Note that;
the two shapes need to be the size to be Congruent i.e. only rotation, reflection and/ or translation is needed.
Remember this:

Two line segments are Congruent if they have the same length.

Two angles are Congruent if they have the same measure.

Two circles are Congruent if they have the same diameter.

When two straight lines intersect, they form four angles.
Each opposite pair are called vertical angles and they are congruent.
Vertical angles are also called opposite angles. See figure below for more understanding:
1 1441011123272
Properties of vertical angles

They are Congruent: vertical angles are always of equal measure i.e. a = b, and c = d.

Sum of vertical angles (all four angles) is 3600 i.e. a + b + c + d =3600

Sum of Adjacent angles (angles from each pair) is 1800 i.e. a + d =1800 ; a + c =1800 ; c + b =1800 ; b + d =1800.

Two Triangles are Congruent if their corresponding sides are equal in length and their corresponding angles are equal in size. The symbol for congruent shapes is ≅
The Conditions for Congruence of Triangles
Determine the conditions for congruence of triangles
The following are conditions for two Triangles to be Congruent:

SSS
(side-Side-Side): if three pairs of sides of two Triangles are equal in
length, then the Triangles are Congruent. Consider example below
showing two Triangles with equal lengths of the corresponding sides.

2 1441011479950
Example 1
Prove that the two Triangles (ΔABC and ΔBCD) below are Congruent.
3 1441011608807
Solution
4 1441011785151
Another condition;

 

SAS
(Side-Angle-Side): This means that we have two Triangles where we know
two sides and the included angles are equal. For example;

5 1441011981245
If
the two sides and the included angle of one Triangle are equal to
corresponding sides and the included angle of the other Triangle, we say
that the two Triangles are Congruent.

ASA
(Angle- Side-Angle): If two angles and the included side of one
Triangle are equal to the two angles and included side of another
Triangle we say that the two Triangles are congruence. For example

6 1441012122659
AAS condition;

AAS
(Angle-Angle-Side): If two angles and non included side of one triangle
are equal to the corresponding angles and non included side of the
other Triangle, then the two triangles are congruent. For example

7 1441012249195

HL
(hypotenuse-Leg): This is applicable only to a right angled triangle.
The longest side of a right angled triangle is called hypotenuse and the
other two sides are legs.

It means we have two right angled triangles with:

The same length of hypotenuse and

The same length for one of the other two legs.

If
the hypotenuse and one leg of one right angled triangle are equal to a
corresponding hypotenuse and one leg of the other right angled triangle,
the two triangles are congruent.
For example
7 1441012249195
Important
note: Do not use AAA (Angle-Angle-Angle). This means we are given all
three angles of a triangle but no sides. This is not enough information
to decide whether the two triangles are congruent or not because the
Triangles can have the same angles but different size. See an
illustration below:
9 1441012556134
The two triangles are not congruent.
Without knowing at least one side, we can’t be sure that the triangles are congruent.
Congruence of Triangle
Prove congruence of triangle
Example 2
Prove that the two Triangles (ΔABC and ΔBCD) below are Congruent.
3 1441011608807
Solution;
4 1441011785151
Theorems on Congruence of Triangles to Solve Related Problems
Apply theorems on congruence of triangles to solve related problems
Isosceles Triangle Theorem
The figure below illustrates an example of an isosceles triangle:
10 1441012697723
An
isosceles triangle has two congruent sides (opposite sides) and two
congruent angles.
The congruent angles are called base angles and the
other angle is called vertex angle.
The angles A and B are base angles
and angle C is the vertex angle.
The base angle Theorem
If two sides of a triangle are congruent, then the angles opposite to these sides are congruent
11 1441013373110
Required to prove: angle A = angle B
12 1441013550120
Therefore, the base angles i.e. angle CAS and angle BCS are equal (by the definition of a congruence of triangles).
It states that, if two angles of a triangle are congruent, then sides opposite those angles are congruent.
13 1441013701420
14 1441013831733
15 1441013893634
Exercise 1
1.
In the isosceles triangle ABC, BA and BC are congruent. D and E are
points on AC such that AD is congruent to BD and BE is congruent to BC.
Show that the triangles ABD and CBE are congruent
16 1441014101837
2. ABCD is a parallelogram and BEFC is a square. Show that triangles ABE and DCF are congruent.
17 1441014243470
3. Use the figure below to answer the following questions:
18 1441014338903
19 1441014479251
20 1441014606199

RELATED ARTICLESMORE FROM AUTHOR

12 COMMENTS

  1. Someone essentially assist to make severely articles I would state.

    This is the first time I frequented your website page and so far?

    I surprised with the analysis you made to make this actual publish incredible.
    Fantastic job!

  2. Simply desire to say your article is as surprising.

    The clarity in your post is simply excellent and i can assume you’re an expert on this subject.
    Fine with your permission let me to grab your feed to keep updated with forthcoming post.
    Thanks a million and please continue the enjoyable work.

  5. With havin so much written content do you ever run into any issues of plagorism
    or copyright violation? My website has a lot of exclusive content I’ve either authored myself or outsourced but it appears a lot of
    it is popping it up all over the web without my agreement.

    Do you know any techniques to help stop content from being ripped
    off? I’d definitely appreciate it.

  7. Hi everyone, it’s my first pay a quick visit at this web site, and post is
    really fruitful in support of me, keep up posting such articles.

  9. My spouse and I stumbled over here by a different page and thought I should
    check things out. I like what I see so now i’m following you.
    Look forward to checking out your web page again.

    Take a look at my web site – xe88 company

  11. Hi, I do believe this is a great blog. I stumbledupon it ;
    ) I am going to revisit once again since i have bookmarked it.

    Money and freedom is the greatest way to change,
    may you be rich and continue to guide others.

  13. Hello! Quick question that’s totally off topic. Do you know how to make
    your site mobile friendly? My website looks weird when viewing from my iphone4.
    I’m trying to find a template or plugin that might be able to resolve this problem.
    If you have any suggestions, please share. Cheers!

  15. Everything is very open with a clear clarification of the challenges.
    It was really informative. Your website is useful.
    Thanks for sharing!

LEAVE A REPLY

Please enter your comment!
Please enter your name here