Angles and Triangles Angles Is a geometrical figure

  • Slides: 18
Download presentation
Angles and Triangles

Angles and Triangles

Angles • Is a geometrical figure that results from the union of two rays.

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

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

Angle Theorems • Two angles are complementary if and only if their sum of

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

Angle Theorems • Two angles are supplementary if and only if their sum of

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

Angle Theorems • Vertical Angles are congruent (Examples: <ACB ≡ <DCE and <ACD ≡

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? ).

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

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

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

Triangles • A triangle is a polygon with 3 vertices and 3 sides. •

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 60 4 60 o A o 40 o D 60 o 4 100 o 6 3 5 50 o 40 o 10 40 o F G 4 I C ∆ABC is an equilateral and an acute triangle. ∆DEF is an isosceles and an obtuse triangle. ∆GHI is a scalene and a right triangle.

Triangle Theorems • The sum of the interior angles of a triangle is always

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

Triangle Theorems • The sum of two of its sides is always greater than

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

Triangle Theorems • For right triangles, the sum of the two angles other than

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

Triangle Theorems • In addition, for right triangles, the measure of the hypotenuse c

Triangle Theorems • In addition, for right triangles, the measure of the hypotenuse c (the longest side of a right triangle) is given by c 2 = a 2 + b 2 (Pythagorean Theorem). B H E 60 4 60 o A o 40 o D 60 o 4 100 o C 6 3 5 50 o 40 o 10 40 o F G 4 I

Triangle Theorems • For isosceles triangles, the measure of the side angles are congruent.

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

Triangle Theorems • For equilateral triangles, the measure of each of its interior angles

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

Triangle Theorems • If given two triangles, if the measure of two of the

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

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 D C E

Similar Triangles A B D C E