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Year 10 Interactive Maths - Second Edition


Congruence

Congruent figures have the same size and the same shape.  The corresponding sides and angles of congruent figures are equal.

Square ABCD and square EFGH.

Square ABCD is congruent to square EFGH as their corresponding sides and angles are equal.

Note:

Congruent figures are exact duplicates of each other.  One could be fitted over the other so that their corresponding parts coincide.

The concept of congruence applies to figures of any type.  In this section we will consider congruent triangles, principles of congruent triangles and their applications.


Congruent Triangles

Congruent triangles have the same size and the same shape. The corresponding sides and the corresponding angles of congruent triangles are equal.

Triangle ABC and DEF have corresponding sides of 3 cm, 4 cm and 5 cm length.  The two triangles also have corresponding angles of size 37 degrees, 53 degrees and 90 degrees.

Triangle ABC and triangle DEF have the same size and shape.  In fact, if you could 'pick up' triangle ABC you could fit it exactly over triangle DEF so that the vertices and sides coincide.  Of course, you would have to put C on top of F, A on D and B on E, because they wouldn't match any other way.  We convey all this information by saying that triangle ABC is congruent to triangle DEF.

In symbolic form, we write this as triangle ABC is congruent to triangle DEF.  The congruent to symbol looks like an equals sign but with three instead of two horizontal lines.

Note:

The order of the points in the names of the triangles is important.  Triangle ABC is congruent to triangle DEF says that the triangles will coincide when A is placed on D, B on E, and C on F.  (We would not say that triangle ABC is congruent to triangle EFD in the above).

The congruent triangles ABC and DEF have equal corresponding sides (AB = DE, BC = EF, CA = FD) and equal corresponding angles (angle A = angle D, angle B = angle E, angle C = angle F).


Principles of Congruent Triangles

The following principles of congruence are used depending on the information given.

1.  The side-side-side (SSS) principle

Two triangles are congruent if corresponding sides are equal.

Triangle ABC and triangle DEF have three pairs of corresponding sides.

2.  The side-angle-side (SAS) principle

Two triangles are congruent if two pairs of corresponding sides and the angle included between the sides are equal.

Triangle ABC and triangle DEF have two pairs of corresponding sides and a pair of corresponding angles.

3.  The angle-side-angle (ASA) principle

Two triangles are congruent if two pairs of corresponding angles and a pair of corresponding sides are equal.

Triangle ABC and triangle DEF have two pairs of corresponding angles and a pair of corresponding sides.

4.  The right angle-hypotenuse-side (RHS) principle

Two right-angled triangles are congruent if the hypotenuses and one pair of corresponding sides are equal.

Triangle ABC and DEF have equal hypotenuses and one pair of corresponding sides that are equal.



Example 4

Find the value of each of the pronumerals in the given pair of triangles. Give reasons for your answers.                        

The two unknown angles in triangle ABC are x degrees and y degrees.

The two known angles in triangle DEF are 41 degrees and 49 degrees.


Solution:

Triangle ABC is congruent to triangle DEF by the RHS principle.  Therefore, x = 41 and y = 49 by the corresponding angles of congruent triangles.


Example 5

Find the value of each of the pronumerals in the given pair of triangles. Give reasons for your answers.

Triangle ABC has one side of length 12 cm and two sides of unknown length in x cm and y cm.

Triangle DEF has one unknown side of length z cm and two known sides of length 9 cm and 6 cm.


Solution:

Triangle ABC is congruent to triangle DEF by the ASA principle.  Therefore, x = 9, y = 6, z = 12 by the corresponding sides of congruent triangles.


Example 6

Find the value of each of the pronumerals in the given pair of triangles. Give reasons for your answers.

Triangle ABC has one unknown angle of size x degrees and two known angles of size 58 degrees and 33 degrees.

Triangle DEF has one known angle of size 89 degrees and two unknown angles of size y degrees and z degrees.


Solution:

Triangle ABC is congruent to triangle DEF by the SSS principle.  Therefore, x = 89, y = 58, z = 33 by the corresponding angles of congruent triangles.


Example 7

Find the value of each of the pronumerals in the given pair of triangles. Give reasons for your answers.

Triangle ABC has two unknown angles of size x degrees and y degrees.

Triangle DEF has two known angles of size 45 degrees and 41 degrees.

Solution:

Triangle ABC is congruent to triangle DFE by the SAS principle.  Therefore, x = 45 and y = 41 by the corresponding angles of the congruent triangles.


Example 8

Use the data in the diagram to prove that triangle ABD is congruent to triangle CDB.

The parallelogram ABCD is divided into two triangles by the line joining BD.  The unknown angles in triangle ABD are a degrees and x degrees.  The unknown angles in triangle BCD are b degrees and y degrees.

Solution:

In triangle ABD and triangle CDB,

a = b   {Alternate angles}.  BD = DB   {Common side}.  x = y   {Alternate angles}.

Therefore, triangle ABD is congruent to triangle CDE by the ASA principle.

As required.


Example 9

In the following diagram, if AD is perpendicular to BC and BD = CD, prove that angle B = angle C.

Triangle ABC is divided into two triangles by the line AD.

Solution:

BD = DC in triangle ABC.

In triangle ABD and triangle ACD,
AD = AD   {Common side}.  Angle ADB = angle ADC = 90 degrees   {Both are right angles}.  BD = CD   {Given}.

Therefore, triangle ABD is congruent to triangle ACD   {SAS}.  Therefore, angle B = angle C   {Corresponding angles of congruent triangles}.
As required.


Key Terms

congruent figures, congruent triangles, corresponding sides, corresponding angles, side-side-side (SSS) principle, side-angle-side (SAS) principle, angle-side-angle (ASA) principle, right angle-hypotenuse-side (RHS) principle


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