Topic 3 - Congruent Triangles and Inequalities

Mrs. Hubbard

scalene

iff none of its sides are equal

isosceles

iff it has at least two equal sides

equilateral

iff all of its sides are equal

obtuse

iff it has an obtuse angle

right

iff it has a right angle

acute

iff all of its angles are acute

equiangular

iff all of its angles are equal

△ABC ≅ △DEF

corresponding parts of congruent triangles are equal

∠A = ∠D; ∠B = ∠E; ∠C = ∠F

AB = DE; BC = EF; AC = DF

two triangles are congruent if there is a correspondence between their vertices such that all of their correspoding sides and angles are equal

△ABC ≅ △DEF; △XYZ ≅ △DEF

△ABC ≅ △XYZ

two triangles congruent to a third triangle are congruent to each other

∠A = ∠D; AB = DE; ∠B = ∠E

△ABC = △DEF

ASA Postulate

if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent

AB = DE; ∠A = ∠D; AC = DF

△ABC = △DEF

SAS Postulate

if teo sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent

AB = BC; ∠C = ∠A

If two sides of a triangle are equal, the angles opposite of theme are equal

∠A = ∠C; BC = AB

if two angles of a triangle are equal, the sides opposite of them are equal

△ABC is equilateral; Triangle ABC is equiangular

An equilateral triangle is equiangular

△ABC is equilangular; Triangle ABC is equilateral

An equiangular triangle is equilateral

△ABC is scalene

If a triangle is scalen, it has no equal sides

AB ≠ BC and AB ≠ AC and BC ≠ AC

A triangle is scalene if it has no equal sides

△ABC is isosceles

If a triangle is isosceles, it has to have at least two equal sides

AB = BC or AB = AC or BC = AC

A triangle is isosceles if it has at least two equal sides

△ABC is equilateral

If a triangle is equilateral, all of its sides are equal

AB = BC = AC

A triangle is equilateral if all of its sides are equal

△ABC is obtuse

If a triangle is obtuse, it has an obtuse angle

∠ABC is obtuse

A triangle is obtuse if it has an obtuse angle

△ABC is right

If a triangle is right, it has a right angle

∠ACB is right

A triangle is right if it has a right angle

△ABC is acute

If a triangle is acute, it has three acute angles

∠A, ∠B, and ∠C are acute angles

A triangle is acute if it has three acute angles

△ABC is equiangular

If a triangle is equiangular, all of its angles are equal

∠A = ∠B = ∠C

A triangle is equiangular if all of its angles are equal

AB = DE; BC = EF; AC = DF | △ABC ≅ △DEF

SSS Theorem

a and b are real numbers | a < b; a = b; a > b

the “three possibilities” property

a < b; b < c | a < c

transitive property of inequality

a < b | a + c < b + c

addition property

a < b | a - c < b - c

subtraction property

a < b and c > 0 | ac < bc

multiplication property

a < b and c > 0 | a/c < b/c

division property

a > b and c > d | a + c > b + d

addition theorem of inequality

a > 0, b > 0, and a + b = c | c > a and c > b

whole greater than part theorem

∠ACD is an exterior angle of △ABC

If an angle is an exterior angle of a triangle, it forms a linear pair with an angle of the triangle

∠ACB and ∠ACD are a linear pair

if an angle forms a linear pair with an angle of a triangle, it is an exterior angle of the triangle

∠ACD is an exterior angle of △ABC | ∠ACD > ∠A and ∠ACD > ∠B

Exterior Angle Theorem

AB < BC < AC | ∠C < ∠A < ∠B

If two sides of a triangle are unequal, the angle opposite them are unequal in the same order

∠A < ∠B < ∠C | BC < AC < AB

If two angles of a triangle are unequal, the sides opposite them are unequal in the same order

△ABC | AB + BC > AC; AB + AC > BC; BC + AC > AB

The Triangle Inequality Theorem