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4-1 Classifying Triangles
I. Geometric ShapesWhat is a triangle? A TRIANGLE is a three-sided polygon.
IV. Classification by Sides: Triangles can also be classified according to the number of congruent sides they have.
4-2 Measuring Angles in Triangles
I. Triangle Angle Sum Theorem:4-1 The sum of the measures of the angles in a triangle is 180.
II. Third Angle Congruence: Theorem 4-2 Third Angle Theorem
If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.
III.Triangle Exterior Angles & its Corollaries:
Theorem 4-3 Exterior Angle Theorem- The measure of an exterior angle of a triangle is equal tothe sum of the measures of the two remote interior angles.
By the way.... what is a corollary? A COROLLARY is a statement that can be easily proved using a theorem..... A better way of saying this... is that a corollary is a fact or statement that directly falls from a given theorem.
Corollary 4-1 -
The acute angles of a right triangle are complementary.
Corollary 4-2 – There can be at most one right or obtuse angle in a triangle
I. When two triangles are congruent to each other then......there are SIX pieces of information that must be true: 3 congruent corresponding sides 3 congruent corresponding angles
4-3 Congruent Triangles
II. Definition of Congruent Triangles (CPCTC)
Two triangles are congruent if and only if their corresponding parts are congruent.
IV. Examples• Triangle RST is isosceles with S as
the vertex angle. If ST = 3x - 11, SR = x + 3, and RT = x - 2, find RT.
3. Given triangle STU with S (2,3), T (4,3) and U (3,-2). Use the distance formula to prove it is isosceles.
Examples
1. Find the value of x.
2. What is the value of
angle W if
angle X is 59 and
angle XYZ is 137?
2. Refer to the design shown. How many of the triangles in the design appear to be congruent to triangle A?
II. Examples
1. PQR with P(3,4) Q (2,2) R (7,2) STU with S(6,-3) U (4,-7) T (4,-2) Prove that PQR SUT
I. Modification of 4-3
Postulate 4-3 AAS (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, the two triangles are congruent.
II. Theorem 4-6 Isosceles Triangle Theorem (ITT)
If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Summary - In other words if you have two congruent sides, you have two congruent base angles.
III. Theorem 4-7 Converse of the ITT
If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Summary - If you have two congruent angles, then you have two congruent legs.