Proofs Using Congruence:Remember congruence is where two shapes have exactly the same side lengths and internal angles and these all correlate/match up, i.e.

Triangle FDE (is congruent to) CAB:

“this is because the angles match up without rearranging them as do the side lengths”

This means congruent shapes can be flipped and rotated as long as they are not warped to be considered congruent:

Q: To prove congruence what do you need to observe?

a) All internal angles are the same.

c) All angles and sides correspond.

b) All sides have equal length.

d) All of the above.

Definitions:A Scalene Triangle has no two sides the same length.

i.e.

An Isosceles Triangle has exactly two sides the same length.i.e.

An Equilateral Triangle has three sides the same length.i.e.

Q: What type of triangle could a right angle triangle be?

a) An equilateral.

c) An isosceles or scalene

b) A scalene.

c) An isosceles

Properties:We use properties such as the internal angle and side length to classify triangles, if we look back at the previous slide we can observe this…

A Scalene triangle has no identifiable property as all internal angles are of a different magnitude.

An isosceles triangle has two angles of equal magnitude.

An equilateral triangle has all angles of equal magnitude.

Q: What must be the magnitude of each angle in an Equilateral Triangle? “remember all three are the same and they add up to a special number”

a) 60o

c) 180o

b) 90o

d) 360o

Proof’s Using Similarity:Remember… similar shapes are those which have the same internal angles, not just this these angles must correlate or be in the same place. Basically similar shapes could be flipped and rotated and when superimposed one will be larger than the other.

The initial shape is known as the “Original” the transformation is the “Image”.

Q: How would a pair of similar shapes differ from a pair of congruent shapes?

a) It would be inside out.

c) The lengths would be different.

b) The angles would be different.

d) They wouldn’t correspond.

Midpoint Theorem:The line that joins the midpoint of two sides of a triangle is parallel to the third side, and is half of the third sides length.

The theorem says that MP will be parallel to BC what does this mean for the interior angles, B to M and C to P?

a) They are opposite

c) They are different

b) They are complimentary

d) They are corresponding

Inverse Theorem:Converse to the midpoint theorem a line drawn through the midpoint of one side of a and parallel to a second side will bisect the third side.

Basically stating proof for the midpoint theorem, if you find the midpoint of one side and draw a line parallel to any other you will bisect the third side of the triangle.

Q: What does bisect mean?

a) Divide into equal parts

c) Cut a line

b) Explore a frog

d) Make a triangle

Intercept Theorem:Very easy when parallel lines make equal intercepts on a transversal an other transversal will have equal intercepts too. Have a look below…

Red is parallel…

From this we can assume using similarity thatDE/BC=AD/AB=AE/AC

Q: What is are the names example of all types of angles present when a line transverses two parallel lines.

a) Co-interior

c) Alternate

b) Corresponding

d) All of the above

Why???The whole idea behind using similarity and congruence to prove these ideas is to gain an understanding into the mathematical process behind them.

As you work through the questions ensure that you refer to the relevant worked examples to see how to set out your working.