BASIC MATHEMATICS-TOPIC 5: CONGRUENCE

 CongruenceI’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.

Angles formed by the intersection of two straight linesWhen
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:

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.

Congruence of TrianglesTwo
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 TrianglesDetermine the conditions for congruence of trianglesThe 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 1Prove that the two Triangles (ΔABC and ΔBCD) below are Congruent.

Solution

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;

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

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

• 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

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.Without knowing at least one side, we can’t be sure that the triangles are congruent.Congruence of TriangleProve congruence of triangleExample 2Prove that the two Triangles (ΔABC and ΔBCD) below are Congruent.

Solution;

Theorems on Congruence of Triangles to Solve Related ProblemsApply theorems on congruence of triangles to solve related problemsIsosceles Triangle TheoremThe figure below illustrates an example of an isosceles triangle:

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 TheoremIf two sides of a triangle are congruent, then the angles opposite to these sides are congruent

Required to prove: angle A = angle B

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.

Exercise 11.
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

2. ABCD is a parallelogram and BEFC is a square. Show that triangles ABE and DCF are congruent.

3. Use the figure below to answer the following questions: