Chapter 4
Converse of Isosceles Base Angles Theorem
- if two angles of a triangle are congruent, then the sides opposite those angles are congruent
Third Angles Theorem
- if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent
Isosceles Base Angles Theorem
- if two sides of a triangle are congruent, then the angles opposite those sides are congruent
scalene triangle
- A scalene triangle is a triangle that has three unequal sides - no equal sides
Angle-Angle-Angle (AAA)
- AAA is not a Postulate nor a theorem for congruence - it is the same shape but not the same size
acute triangle
- An acute triangle is a triangle with all three angles acute (less than 90°).
equiangular triangle
- An equiangular triangle is a triangle where all three interior angles are equal in measure. - Because the interior angles of any triangle always add up to 180°, each angle is always a third of that, or 60° - 3 angles equal
obtuse triangle
- An obtuse triangle is one with one obtuse angle (greater than 90°) and two acute angles. - Since a triangle's angles must sum to 180°, no triangle can have more than one obtuse angle.
CPCTC - Corresponding Parts of Congruent Triangles are Congruent
- CPCTC states that if two or more triangles are congruent, then all of their corresponding angles and sides are congruent as well.
corresponding sides
- Corresponding sides and angles are a pair of matching angles or sides that are in the same spot in two different shapes. - the sides with the same number of arcs are congruent - Note: These shapes must either be similar or congruent.
congruent figures
- Figures that have the exact same shape and size are called congruent figures. - same size and shape - that is, if their corresponding angles and sides are equal. - make sure to list the congruent angles in the same order
corresponding angles
- If the measures of the corresponding sides of two triangles are proportional then the triangles are similar. - the angles with the same number of arcs are congruent
right triangle
- Right triangles are triangles in which one of the interior angles is 90o. - The side opposite of the right angle is called the hypotenuse. - The sides adjacent to the right angle are the legs. - When using the Pythagorean Theorem, the hypotenuse or its length is often labeled with a lower case c.
triangle
- a triangle is formed by 3 non-collinear points connected by segments
equilateral triangle
- an equilateral triangle is a triangle in which all three sides are equal. - is ALWAYS isosceles triangle - 3 sides equal
isosceles triangle
- an isosceles triangle is a triangle that has two sides of equal length. - it may or may not be equilateral
exterior angle
- angles that are adjacent to the interior angles - two sides of the angle are formed by an extension of a side of the triangle AND a side of the triangle
Perpendicular Bisector of the Base of an Isosceles Triangle
- if a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base
If Equiangular then Equilateral Triangle Theorem
- if a triangle is equiangular, then the triangle is equilateral - ALL triangle equiangular are also equilateral
If Equilateral then Equiangular Triangle Theorem
- if a triangle is equilateral, then the triangle is equiangular - ALL equilateral triangle are also equiangular
Hypotenuse-Leg (H-L) Theorem
- if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent
Side-Side-Side (SSS) Postulate
- if the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent
Angle-Angle-Side (AAS) Theorem
- if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. - if two pair of corresponding angles are congruent, then the third pair is congruent (Third Angles Theorem).
Angle-Side-Angle (ASA) Postulate
- if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
Side-Angle-Side (SAS) Postulate
- if two side and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent - Postulate does not refer to any two sides and any one angle. Rather SAS refers to two sides and the only included angle of the two sides.
hypotenuse
- in a right triangle, the side opposite the right angle - longest side in the triangle
legs of an isosceles triangle
- isosceles triangle has exactly two congruent (equal measure) sides, then these two sides are its legs
vertex
- non-collinear points
sides of a triangle
- segments joining the points
corollary
- special name given to a theorem that is easy to prove as a direct result of another previously proved theorem - it is a theorem
vertex angle of an isosceles triangle
- the angle opposite the base is the vertex angle
Exterior Angle of a Triangle Corollary
- the measure of each exterior angle of a triangle equals the sum of the measures of its two non-adjacent interior angles
base of an isosceles triangle
- the third side is the base
base angles of an isosceles triangle
- the two angles adjacent to the base are the base angles
legs of a right triangle
- the two sides (segments) connecting to the 90 degree angle
NO SSA (side-side-angle)
- there is not SSA congruence postulate or theorem for triangles in general
interior angle
- three original angles of a triangle
Acute Angles of a Right Triangle Corollary
- two acute angles of a right triangle are complementary
Conditions for Hypotenuse-Leg (H-L) Theorem
3 Conditions: 1. there are two right triangles 2. triangles have congruent hypotenuses 3. the is at least one pair of congruent legs
Triangle Facts
With the Theorems presented you can conclude: equilateral = equiangular = a triangle with all angles of 60 degrees