Geometry – Chapter 4 Congruent Triangles

4.1 – Apply Angle Sum Properties Triangle Polygon with three sides & three vertices Triangles can be classified by side and angles Classifying Triangles by Sides Scalene Iscoceles Equilateral No congruent sides Two congruent sides All sides congruent Classifying Triangles by Angles Acute Right Obtuse Equiangular 3 acute angles (< 90) 1 right angle (= 90) 1 obtuse angle (> 90) 3 congruent angles

Example 2 Classify ∆PQO by its sides, then determine if the triangle is right. Points are: P (-1, 2) Q (6, 3) O (0, 0) GP: #1-2

Angles Interior Angles Exterior Angles Angles on the inside of the triangle (there are three) Exterior Angles Angles that form linear pairs with interior angles (there are 6)

Theorems 4.1 – Triangle Sum Theorem 4.2 – Exterior Angle Theorem The sum of the measure of the interior angles of a triangle is 180° 4.2 – Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles Example #3 p. 209

Corollary to a theorem Corollary to a theorem Statement that can be proved easily using the theorem Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary Example 4 A tiled staircase forms a right triangle. The measure of one acute angle in the triangle is twice the measure of the other. Find the measure of each acute angle GP #3 & 5 p. 210

4.2 – Apply congruence & triangles Two geometric figures are congruent if they have exactly the same size and shape Congruent figures All parts of one figure are congruent to the corresponding parts of the other figure (corresponding sides & corresponding angles) Congruence Statements Be sure to name figures by their corresponding vertices!

examples Example 1 Example 2 GP #1-3 p. 216 Writing a congruence statement and identifying all congruent parts Example 2 Using properties of congruent figures 𝐷𝐸𝐹𝐺≅𝑆𝑃𝑄𝑅 GP #1-3 p. 216

Third angles theorem Theorem 4.3 – Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent

Using third angles theorem Example 4 Find m< BDC A B 45° N 30° C D GP #4-5 p. 217

Properties of congruent triangles The properties of congruence that are true for segments and angles are also true for triangles Theorem 4.4 – Properties of Congruent Triangles Reflexive property ∆𝐴𝐵𝐶≅∆𝐴𝐵𝐶 Symmetric property ∆𝐴𝐵𝐶≅∆𝐷𝐸𝐹, 𝑡ℎ𝑒𝑛 ∆𝐷𝐸𝐹≅∆𝐴𝐵𝐶 Transitive property ∆𝐴𝐵𝐶≅∆𝐷𝐸𝐹 𝑎𝑛𝑑 ∆𝐷𝐸𝐹≅∆𝑋𝑌𝑍 𝑡ℎ𝑒𝑛 ∆𝐴𝐵𝐶≅∆𝑋𝑌𝑍

4.3 – relate transformations & congruence Rigid motion Transformation that preserves length, angle measure, and area Examples of rigid motions (isometry): translations, reflections, rotations Congruent figures and Transformations Two figures are congruent if and only if one or more rigid motions can be used to move one figure onto the other. If any combination of translations, reflections, and rotations can be used to move one shape onto the other, the figures are congruent

4.4 – Prove triangles congruent by SSS Postulate 19 – Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent

Example 1 Use the SSS congruence postulate GP #1-3 p. 232

4.5 – congruence by SAS and HL Postulate 20 – Side-Angle-Side (SAS) Congruence Postulate 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

Right triangles In a right triangle, the sides adjacent to the right angles are called the legs The side opposite the right angle is called the hypotenuse Theorem 4.5 – Hypotenuse-Leg (HL) Congruence Theorem 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

4.6 – Prove using ASA & AAS Postulate 21 – Angle-Side-Angle (ASA) Congruence Postulate 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

AAS theorem Theorem 4.6 – Angle-Angle-Side (AAS) Congruence Theorem 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

Triangles postulates &theorems 5 methods for proving that triangles are congruent SSS SAS HL (right angles only) ASA AAS All 3 sides are congruent Two sides and the included angle are congruent Hypotenuse and one of the legs are congruent Two angles and the included side are congruent Two angles and a non-included side are congruent

4.7 – use congruent triangles Congruent triangles have congruent corresponding parts If two triangles are congruent, their corresponding parts must be congruent as well

Euclid’s river example

4.8 – use isosceles and equilateral triangles Legs Two congruent sides of an isosceles triangle Vertex angle Angle formed by the legs Base Third side of an isosceles triangle Base angles Angles adjacent to the base (opposite the legs)

Isosceles triangles theorem Theorem 4.7 – Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent Theorem 4.8 – Converse of Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent

Example 1 Name two congruent angles F D E GP #1-2 p. 264

Corollaries Corollary to the Base Angles Theorem If a triangle is equilateral, then it is equiangular Corollary to the Converse of Base Angles Theorem If a triangle is equiangular, then it is equilateral Example 2 If a triangle is equilateral, what is the measure of each angle? Example 3 – on board GP #5 p. 266