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CHAPTER FOUR
 Triangle is a figure formed by three segments joining three noncollinear points.
 The symbol for a triangle is .
 Vertex is the point joining the sides of a triangle.
 To name a triangle, put the symbol for a triangle, , followed in
clockwise order the capital letters naming the points at each vertex.
 A triangle can be classified either by its sides or its angles.
 Side Classification of a Triangle
o Equilateral is when all three sides are congruent.
o Isosceles is when two sides are congruent. The word
“isosceles” means two legs the same.
o Scalene is when no two sides are congruent. The word
“scalene” means odd.
 An equal number of slashes (hatch marks or tally marks) on the sides
of a triangle indicate that those sides are congruent.
 Angle Classification of a Triangle
o Acute is when all angles are less than 90 degrees in measure.
o Equiangular is when all angles are congruent. An equiangular
triangle is also an Acute triangle.
o Right is when one angle of the triangle measures exactly 90
degrees.
o Obtuse is when one angle of the triangle measures more than 90
degrees.
 Adjacent Sides are two sides sharing a common vertex.
 Right Triangle parts have special names. The side opposite the right
angle is called the hypotenuse. The other two sides are called legs.
 Isosceles Triangle parts have special names. The two congruent sides
are called legs. The bottom of the triangle is called the base.
 Base Angles are the angles formed by the base and the legs of an
Isosceles Triangle.
 Every triangle can be classified in two ways by its sides and by its
angles. There are seven possible side/angle classifications for
triangles.
o Scalene / Acute
o Isosceles / Acute
o Equilateral / Acute
o Scalene / Right
o Isosceles / Right
o Scalene / Obtuse
o Isosceles / Obtuse
 Interior Angles are the angles inside the triangle.
 Exterior Angles are the angles formed by extending the sides of the
triangle.
 Triangle Sum Theorem states that the sum of the measures of the
interior angles of a triangle is 180 degrees.
 An equiangular triangle has three equal angle measures. Since all
interior angles of any triangle add up to 180 degrees, then each of the
three angles measures 60 degrees.
 A right triangle has one angle that measures exactly 90 degrees. Since
all interior angles of any triangle add up to 180 degrees, then neither
of the other two angles can be equal to or larger than 90 degrees.
Therefore, in a right triangle to two angles that are not right must be
acute or less than 90 degrees.
 Exterior Angle Theorem states that the measure of an exterior angle of
a triangle is equal to the sum of the measure of the two non-adjacent
interior angles.
 Corollary to a Theorem is a statement that can be proved easily using
the theorem.
 Corollary to the Triangle Sum Theorem states that the acute angles of
a right triangle are complementary.
 Congruent means that the two figures are the same shape and size.
 Corresponding Angles are angles that are located in the same position
but on two different shapes of equal type.
 Corresponding Sides are sides that are located in the same position
but on two different shapes of equal type.
 If two triangles are congruent then their Corresponding Angles are
congruent.
 If two triangles are congruent then their Corresponding Sides are
congruent.
 A congruence statement of two triangles is stated as ABC  DEF.
 Corresponding does not mean congruent.
 Single, double and triple arcs are used to show congruent angles.
 Third Angle Theorem states that if two angles of one triangle are
congruent to two angles of another triangle, then the third angles are
also congruent.
 Reflexive Property of Congruent Triangles states that every triangle is
congruent to itself.
 Symmetric Property of Congruent Triangles states that if the first
triangle is congruent to the second triangle, then the second triangle is
congruent to the first triangle.
 Transitive Property of Congruent Triangles states that if the first
triangle is congruent to the second triangle and the second triangle is
congruent to a third triangle, then the first triangle is congruent to the
third triangle.
 Note that two triangles do not have to have the same orientation in
order to be congruent.
 In congruent triangles, congruent sides are opposite congruent angles.
 Always list the names of congruent triangles in the order of
corresponding parts.
 There are six ways to name each pair of congruent triangles.
 Side-Side-Side (SSS) Congruence Postulate states that if three sides of
one triangle are congruent to three sides of a second triangle, then the
two triangles are congruent.
 Side-Angle-Side (SAS) Congruence Postulate states that if two sides
and the included angle of one triangle are congruent to two sides and
the included angle of a second triangle, then the two triangles are
congruent.
 Included Angle is the interior angle formed where two sides meet.
The letter designating the angle also appears in the anme of both
segments that form the angle.
 Corresponding Parts of Congruent Triangles are Congruent states
that once you have established that two triangles are congruent then
you can state that each corresponding part is congruent.
 Distance Formula
(x1 – x2)2 + (y1 – y2)2
 Angle-Side-Angle (ASA) Congruence Postulate states that if two
angles and the included side of one triangle are congruent to two
angles and the included side of a second triangle, then the two
triangles are congruent.
 Angle-Angle-Side (AAS) Congruence Postulate states that if two
angles and a non-included side of one triangle are congruent to two
angles and the corresponding non-included side of a second triangle,
then the two triangles are congruent.
 Included Side is the side between two angles of a triangle. It is the
side that each of the angles share.
 When triangles overlap, it is a good idea to draw each triangle
separately and label the congruent parts.
 Isosceles Triangle is a triangle in which two sides of the triangle are
congruent. Isosceles means “two legs the same”.
 Equilateral Triangle is a triangle in which all sides and all angles are
congruent.
 Right Triangle is a triangle which has one 90 degree angle (right
angle).
 Base Angles are the two angles adjacent to the base of an isosceles
triangle.
 Vertex Angle is the angle opposite the base in an isosceles triangle.
 Base Angles Theorem states that if two sides of a triangle are
congruent, then the angles opposite them are congruent.
 Converse of the Base Angles Theorem states that if two angles of a
triangle are congruent, then the sides opposite them are congruent.
 Corollary to Base Angles Theorem states that if a triangle is
equilateral, then its is equiangular.
 Corollary to Converse of the Base Angles Theorem states that if a
triangle is equiangular, then it is equilateral.
 Hypotenuse-Leg (HL) Congruence Theorem states that 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.
 Could you use the HL Congruence Theorem to prove two isosceles
triangles are congruent? (Yes but only if they are isosceles right
triangles.)
 Hypotenuse Acute Angle (HA) Theorem states that if the hypotensue
and an acute angle of one right triangle are congruent to the
hypotenuse and corresponding angle of another right triangle, then the
triangles are congruent. (NOTE: Not in textbook)
 Leg Acute Angle (LA) Theorem states that if one leg and an acute
angle of a right triangle are congruent to the corresponding leg and
angle of another right triangle, then the triangles are congruent.
(NOTE: Not in textbook)
 Leg-Leg (LL) Theorem states that if two legs of one right triangle are
congruent to the corresponding legs of another right triangle, then the
triangles are congruent. (NOTE: Not in textbook)
 The Hypotenuse- Leg (HL), Hypotenuse Acute Angle (HA), Leg Acute
Angle (LA), and Leg-Leg (LL) Theorems can only be used with right
triangles.
 Remember the Hypotenuse is always the longest side and is always
opposite the right angle.
 Coordinate Proof is a proof in which you place geometric figures in a
coordinate plane, then use the Distance and Midpoint Formulas as
well as congruence theorems to prove statements about the figures.