**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.

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.

- 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.

**Angles formed by the intersection of two straight lines**

**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 360
^{0}i.e. a + b + c + d =360^{0} - Sum of Adjacent angles (angles from each pair) is 180
^{0}i.e. a + d =180^{0}; a + c =180^{0}; c + b =180^{0}; b + d =180^{0}.

**Congruence of Triangles**

The symbol for congruent shapes is **≅**

**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.

**Example 1**

**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;

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

- 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

- 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.

- The same length of hypotenuse and
- The same length for one of the other two legs.

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

**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:

**The two triangles are not congruent.**

Example 2

Prove that the two Triangles (ΔABC and ΔBCD) below are Congruent.

**Isosceles Triangle Theorem**

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**

**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**

Really enjoyed this blog article. Really Cool.

Thanks so much for the article post.Thanks Again.

Very good post. I’m going through many of these issues as well..

cymbalta withdrawal help duloxetine and addison’s disease

I wanted to thank you for this good read!! I certainly enjoyed every little bit of it. I have got you book-marked to check out new things you post…

Howdy! I know this is kinda off topic but I was wonderingif you knew where I could locate a captcha plugin for mycomment form? I’m using the same blog platform as yours and I’mhaving problems finding one? Thanks a lot!

Really enjoyed this post. Will read on…

A big thank you for your article post.Much thanks again. Will read on…

Hi there! This post could not be written much better! Looking through this post reminds me of my previous roommate! He continually kept talking about this. I’ll send this post to him. Fairly certain he’ll have a great read. Many thanks for sharing!

After I initially commented I clicked the -Notify me when new feedback are added- checkbox and now every time a comment is added I get 4 emails with the same comment. Is there any way you’ll be able to remove me from that service? Thanks!

Hi! I could have sworn I’ve been to this blog before but after checking through some of the post I realized it’s new to me. Anyways, I’m definitely delighted I found it and I’ll be book-marking and checking back often!

Thanks for sharing, this is a fantastic blog article.Much thanks again. Keep writing.

Very nice post. I certainly appreciate this website. Keep writing!