4.1 – Apply triangle sum properties Geometry Chapter 4 4.1 – Apply triangle sum properties

Apply Triangle Sum Properties Objective: Students will be able to classify triangles, and use those classifications to find the measures of their angles. Agenda Classifying Triangles (By Sides and Angles) Triangle Sum Exterior Angles

Triangle Properties Each of the three points of a triangles is known as a vertex From ∆𝐴𝐵𝐶, we can see that Vertices: points A, B, and C. Sides: 𝐴𝐵 , 𝐵𝐶 , 𝐶𝐴 Angles: <𝐴, <𝐵, <𝐶 A B C

Types of Triangles A triangle can be classified by the number of congruent sides it has Scalene Triangle Equilateral Triangle Isosceles Triangle No Sides Congruent At least Two Sides Congruent All Sides Congruent

Types of Triangles A triangle can also be classified by the angles present in them Acute Triangle Obtuse Triangle 3 Acute <‘s 1 obtuse < Right Triangle Equiangular Triangle 1 Right < All Congruent <‘s

Types of Triangles Example 1: Classify the following triangles based off their side lengths and/or angle measures. 𝟖 𝟓 𝟕 𝟓 𝟑 𝟗𝟐° 𝟑𝟕° 𝟕𝟕° 𝟔𝟒°

Angles of a Triangle – Interior Angles When the sides of a triangle are extended, other angles are created. The original angles of the triangle are called the interior angles.

The Sum of the Angles of a Triangle Theorem 4.1 – Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180. A B C 𝑚<𝐴+𝑚<𝐵+𝑚<𝐶=180°

The Sum of the Angles of a Triangle Example 2: Find the value of x.

The Sum of the Angles of a Triangle Example 2: Find the value of x. We can apply theorem 4.1: 57+𝑥+2𝑥=180 57+3𝑥=180 3𝑥=123 𝒙=𝟒𝟏

The Sum of the Angles of a Triangle Find the value of x. We can apply theorem 4.1: 𝑥+𝑥+10+87=180 97+2𝑥=180 2𝑥=83 𝒙=𝟒𝟏.𝟓 𝟖𝟕° 𝒙° (𝒙+𝟏𝟎)°

Corollaries A statement that can be proved easily by applying a theorem is often called a corollary of the theorem. The following statement is a corollary of theorem 4-1 (The Triangle Sum Theorem):

Corollaries B A C 𝑚<𝐴+𝑚<𝐵=90° A statement that can be proved easily by applying a theorem is often called a corollary of the theorem. The following statement is a corollary of theorem 4-1 (The Triangle Sum Theorem): Corollary: The acute angles of a right triangle are complementary. B A C 𝑚<𝐴+𝑚<𝐵=90°

Corollaries Example 3: Find the value of x, then find 𝒎<𝑨 and 𝒎<𝑩. B A C 𝑥° 2𝑥°

Corollaries Example 3: Find the value of x, then find 𝒎<𝑨 and 𝒎<𝑩. We can use the corollary: 𝑥+2𝑥=90 3𝑥=90 𝒙=𝟑𝟎° B A C 𝑥° 2𝑥°

Corollaries Example 3: Find the value of x, then find 𝒎<𝑨 and 𝒎<𝑩. For 𝒎<𝑨: 𝑚<𝐴=𝑥=30° B A C 𝑥° 2𝑥° For 𝒎<𝑩: 𝑚<𝐵=2𝑥=60°

Corollaries Find the value of x, then find 𝒎<𝑨 and 𝒎<𝑩. We can use the corollary: 𝑥+𝑥=90 2𝑥=90 𝒙=𝟒𝟓° B A C 𝑥° For 𝒎<𝑨: 𝑚<𝐴=𝑥=45° For 𝒎<𝑩: 𝑚<𝐵=𝑥=45°

Angles of a Triangle – Interior Angles When the sides of a triangle are extended, other angles are created. The angles created by the extended lines, outside the triangle, are called exterior angles. These angles form a linear pair with an interior angle of the triangle.

Exterior Angles Continued Theorem 4.2 – Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles. 𝟏 A B C 𝑚<1=𝑚<𝐴+𝑚<𝐵

Exterior Angles Example 4: Find the value of x. Then find the measure of the exterior angle.

Exterior Angles Example 4: Find the value of x. Then find the measure of the exterior angle. We can apply theorem 4.2: 5𝑥=120+𝑥 4𝑥=120 𝒙=𝟑𝟎

Exterior Angles Example 4: Find the value of x. Then find the measure of the exterior angle. For The Exterior Angle: 5𝑥=5(30) =𝟏𝟓𝟎

Exterior Angles Find the value of x. Then find 𝑚<𝐽𝐾𝑀 We can apply theorem 4.2: 𝑥+70=2𝑥−5 𝒙=𝟕𝟓 𝟕𝟎° 𝑲 𝑱 𝑳 𝒙° (𝟐𝒙−𝟓)° 𝑴 For 𝒎<𝑱𝑲𝑴: 𝑚<𝐽𝐾𝑀=2 75 −5 =150−5=𝟏𝟒𝟓

𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥. 𝟖𝟓° 𝒙° (𝟐𝒙−𝟏𝟓)° 1.) 2.) 𝟏𝟑𝟔° 𝒙° 𝟑𝒙°