Triangle Congruence

AAS Postulate (Angle-Angle-Side)

if two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent

Addition Property of Equality

A proof used when you are using addition to point out an equality (ex: <3+<4= <5+<6)

Reflexive Property

A quantity is congruent (equal) to itself. a = a (Can happen to angles, segments and lines )

What are the methods that don't always work?

AAA ( Angle, Angle Angle ) and SSA/ASS (Side Side Angle/Angle Side Side)

SAS Postulate (Side-Angle-Side)

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Base Angle Theorem (Isosceles Triangle)

If two sides of a triangle are congruent, the angles opposite these sides are congruent.

Whether it is a perpendicular bisector, midpoint or just a general bisector.. remember:

If you are using any of these terms as a definition proof, you must name them correctly. Don't call a bisector a perpendicular bisector.

What has to be in order when you do triangle proofs?

Only the names of a triangle are to be ordered, the names of angles and segments do not have to be lined up

substition property

if x = y, then x can be substituted in for y in any equation, and y can be substituted for x in any equation.

When you have parallel lines, remember..

make sure to observe all congruent angles involved

Hypotenuse

the side of a right triangle opposite the right angle

A radius of a circle are all

congruent

Angle Addition Postulate

for any angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts

Midpoint

A point that divides a segment into two congruent segments

isoceles triangle

a triangle that has 2 congruent sides

Tip on 'perpendicular'

A perpendicular does not bisect, it simply creates a 90 angle

SSA (side side angle)

"SSA" is when we know two sides and an angle that is not the angle between the sides.

diagonal

(geometry) a straight line connecting any two vertices of a polygon that are not adjacent

congruent triangles

2 triangles are congruent if and only if all pairs of corresponding sides and angles are congruent

AAA (Angle-Angle-Angle)

3 angles that are congruent. not enough information to prove the triangles are congruent because we don't know the sides.

What is true about all rectangles?

All opposite sides of a rectangle are congruent

CPOCFAC

Corresponding parts of congruent figures are congruent. ( Can only be used after the triangle has been proved congruent! It is used with proving a figure of an already congruent triangle is congruent to the corresponding triangles figure )

Segment Addition Postulate

If B is between A and C, then AB + BC = AC

Subtraction Property of Equality

If a=b, then a-c=b-c

Substitution

If a=b, then b can replace a in any expression. This can be used when showing the full value of an addition property of equality ex: <3+<4= <5+<6 is the addition property, but with substitution.. it becomes <10 = <11 with the mindset that the first angles are equal to 10 and the other angles is equal to 11.

Hypotenuse-Leg Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

SSS Postulate (Side-Side-Side)

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

ASA Postulate (Angle-Side-Angle)

If two angles and the included side of one triangle are congruent

Common proofs for triangle congruence:

Reflexive Property, Given, Vertical, same side interior angles(only with parallel lines), subtraction, base angle theorem, segment addition postulate, linear pair, definition of bisector or perpendicular bisector, opposite sides are congruent on a rectangle(only when rectangle is stated), definition of midpoint, substitution, triangle = 180

Whenever you see two sides of a triangle are congruent that is given..

Remember the base angle theorem and write down your observation from this

The methods to finding triangle congruence are:

SSS, SAS, ASA, HL, AAS

Triangle congruence letters are only used to prove:

TRIANGLE CONGRUENCE (EX: triangleABC triangle= JNH. Don't use for any angles)

included angle

The angle made by two lines with a common vertex

Radius

The distance from the center of a circle to any point on the circle

Just because a line is bisecting another line does not mean the line bisecting is being bisected as well.

The line that is bisected has been cut in half. But the line that has bisected this line is not cut in half unless both lines bisect each other.

What do the lines on the side of the triangle mean?

This is to show which parts of a shape are congruent to others, these can be used on two corresponding parts of congruent shapes. There can also be a circular shape on the angles, this is to identify congruency with the angles.

How to find what parts of a triangle line up?

You can do this by looking at obvious points and going from there or you can physically use your brain to picture the shape combining to fit into the other shape

How can you find triangle congruence?

You can logically find triangle congruence by indiviually finding and comparing each angle and each side length. But, you can also use a few methods