Angles and Triangles

Angles Is a geometrical figure that results from the union of two rays. The point of union of the two rays is called the vertex. The rays themselves are called sides. Side Vertex Side

Naming Angles Can be named either by the names of one of the points in each side and the name of the vertex, the name of the vertex, or by a Greek letter. A ᶿ A B C <A <ABC

Angle Theorems Two angles are complementary if and only if their sum of their angle measures is 90o = 90o + 50o 40o B A

Angle Theorems Two angles are supplementary if and only if their sum of their angle measures is 180o = + 140o 180o 40o A B

Angle Theorems Vertical Angles are congruent (Examples: <ACB ≡ <DCE and <ACD ≡ <BCE). A D C B E

Angle Theorems Linear Angles are supplementary (What is the measure of <DBC?). A D 130o B C

Finding Angle Measures in Parallel Lines C A B D E G H F

Triangles A triangle is a polygon with 3 vertices and 3 sides. Triangles can be classified either according to the measure of its largest angle, or according to the measurements of its sides. B H E 5 50o 60o 6 6 100o 3 4 4 40o 40o 40o D F G 60o 60o 10 4 I A C 4 ∆GHI is a scalene and a right triangle. ∆DEF is an isosceles and an obtuse triangle. ∆ABC is an equilateral and an acute triangle.

Triangle Theorems The sum of the interior angles of a triangle is always 180o. B H E 5 50o 60o 6 6 100o 3 4 4 40o 40o 40o D F G 4 I 60o 60o 10 A C 4

Triangle Theorems The sum of two of its sides is always greater than the other side (Triangle Inequality Theorem). B H E 5 50o 60o 6 6 100o 3 4 4 40o 40o 40o D F G 4 I 60o 60o 10 A C 4

Triangle Theorems For right triangles, the sum of the two angles other than the right angle is always 90o. B H E 5 50o 60o 6 6 100o 3 4 4 40o 40o 40o D F G 4 I 60o 60o 10 A C 4

Triangle Theorems In addition, for right triangles, the measure of the hypotenuse c (the longest side of a right triangle) is given by c2 = a2 + b2 (Pythagorean Theorem). B H E 5 50o 60o 6 6 100o 3 4 4 40o 40o 40o D F G 4 I 60o 60o 10 A C 4

Triangle Theorems For isosceles triangles, the measure of the side angles are congruent. B H E 5 50o 60o 6 6 100o 3 4 4 40o 40o 40o D F G 4 I 60o 60o 10 A C 4

Triangle Theorems For equilateral triangles, the measure of each of its interior angles is always 60o. B H E 5 50o 60o 6 6 100o 3 4 4 40o 40o 40o D F G 4 I 60o 60o 10 A C 4

Triangle Theorems If given two triangles, if the measure of two of the angles and their included side of a triangle are congruent to two of the angles and their included side of another triangle, then they are congruent (ASA) If given two triangles, if the measure of two of the angles and their opposite side of a triangle are congruent to two of the angles and their opposite side of another triangle, then they are congruent (AAS)

Triangle Theorems If given two triangles, if two sides and their included angle of one triangle are congruent to two sides and their included angle of another triangle, then they are congruent (SAS). If given two triangles, if all the sides of one triangle are congruent to all the sides of another triangle, then they are congruent (SSS).

Similar Triangles A B C D E