- In maths, two shapes are
**congruent**if they are**identical**in shape and size- One may be a
**reflection**,**rotation**, or**translation**of the other

- One may be a
- If one shape is an enlargement of the other, then they are not identical in size and so are
**not**congruent

- To show that two shapes are congruent you need to show that they are both the
**same shape**and the**same size** - You do not need to show that they are facing in the same direction
- If a shape has been reflected, rotated or translated, then its image is congruent to it

- Tracing paper can help in the exam if you are unsure whether two shapes are congruent
- Trace over one shape and then see if it fits on top of the other
- Only do this if the image is drawn to scale

- Two triangles are
**congruent**if they are the same**size**and**shape**- Although they may be reflections, translations or rotations of each other

- All three angles and all three sides must be the same in both triangles

- Happily we don’t have to show that all 3 sides and all 3 angles are the same in each triangle
- We only need to show that
**3 of the 6 things**are the same- as long as they are the right three!

- To do this we must use
**one**of the**5 standard tests****SAS**stands for**side, angle, side**If two sides and the angle between them can be shown to be the same, then the triangles are congruent**ASA**stands for**angle, side, angle**If two angles and the side between them can be shown to be the same, then the triangles are congruent**AAS**stands for**angle, angle, side**If two angles and one side can be shown to be the same, then the triangles are congruent**SSS**stands for**side, side, side**If all three sides can be shown to be the same, then the triangles are congruent**RHS**stands for**right angle, hypotenuse, side**Two right-angled triangles can be shown to be congruent if their hypotenuses (the longest side, opposite the right angle) and one other side can be shown to be the same

- To find equivalent angles or sides you may need to use a variety of skills
- eg Parallel Lines, Symmetry, Isosceles Triangles, Circle Theorems, etc

** **

In the diagram below, *A, B*, *C *and *D *are four points on a circle.

*X* is the point of intersection of the lines *AC *and *BD*.

*M *is the midpoint of *AD*.

*XM *is perpendicular to *AD*.

Prove that triangles *AXB *and *DXC *are congruent.

Congruent triangles have three equal angles and three equal sides.

You do not need to show that all of them are equal to prove congruence, you can look for one of the five standard tests of congruence. These are SAS, SSS, ASA, AAS, RHS.

Use the information given in the question to look for corresponding angles and sides that are equal.

Identify any corresponding sides that are equal between triangles *AXB* and *DXC*.

As *M* is the midpoint of *AD,*
*AM *= *MD.
*Triangles

The sides *AX *and *DX *are equal.

Triangles *AXM *and *CXM *are congruent

Identify any corresponding angles that are equal between triangles
*AXB* and *DXC*.

Vertically opposite angles are equal.

Angle *AXB *= angle *DXC *

Vertically opposite angles

Angles in the same segment are equal.

Angle *ABX *= angle *DCX *

Angles subtended from the same arc are equal

The statement that if two corresponding angles and one side are the same then the two triangles are congruent must be made. This is the AAS property (angle, angle, side)

Write which angles or sides are equal out clearly in the proof, making sure to give the reasons why they are equal.

**
AX = XD as triangles AMX and DMX are congruent.
**

Angle

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