Chapter 4 Triangle Congruence By: Maya Richards 5 th Period Geometry

Section 4-1: Congruence and Transformations Transformations: Transformations: Translations – slides Translations – slides Reflections – flips Reflections – flips Rotations – turns Rotations – turns Dilations – gets bigger or smaller (only one that changes size) Dilations – gets bigger or smaller (only one that changes size)

Rotation of 180 degrees around the point (-0.5, -0.5) Translation 6 units right and 2 units up. Dilation of 2x. Reflection across the y-axis.

Section 4-2: Classifying Triangles

Example 1 30° 60° Classify each triangle by its angle measures. A.Triangle EHG Angle EHG is a right angle, so triangle EHG is a right triangle. B. Triangle EFH Angle EFH and angle HFG form a linear pair, so they are supplementary. Therefore measure of angle EFH + measure of angle HFG = 180°. By substitution, measure of angle EFH + 60° = 180°. So measure of angle EFH = 120°. Triangle EFH is an obtuse triangle by definition. 120°

Example Classify each triangle by its side lengths. A.Triangle ABC From the figure, AB is congruent to AC. So AC = 15, and triangle ABC is equilateral. B. Triangle ABD By the Segment Addition Postulate, BD = BC + CD = = 20. Since no sides are congruent, triangle ABD is scalene.

Section 4-3: Angle Relationships in Triangles Triangle Sum Theorem The sum of the angle measures of a triangle is 180 degrees. angle A + angle B + angle C = 180°

Angle 4 is an exterior angle. Its remote interior angles are angle 1 and angle 2. Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Measure of angle 4 = measure of angle 1 + measure of angle 2.

Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. Angle N is congruent to angle T L R N T M S

Example 1

Section 4-4: Congruent Triangles Corresponding Sides AB is congruent to DE BC is congruent to EF AC is congruent to DF Corresponding Angles A is congruent to D B is congruent to E C is congruent to F

Section 4-5: Triangle Congruence: SSS and SAS Side-Side-Side Congruence (SSS) If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Side-Angle-Side Congruence (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Example 1 Use SSS to explain why triangle PQR is congruent to triangle PSR. It is given that PQ is congruent to PS and that QR is congruent to SR. By the Reflexive Property of Congruence, PR is congruent to PR. Therefore triangle PQR is congruent to triangle PSR by SSS.

Section 4-6: Triangle Congruence: ASA, AAS, and HL Angle-Side-Angle Congruence (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Angle-Angle-Side Congruence (AAS) If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent.

Hypotenuse-Leg Congruence (Hy-Leg) If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another triangle, then the triangles are congruent.

Example 1

Section 4-7: Triangle Congruence: CPCTC CPCTC = (“Corresponding Parts of Congruent Triangles are Congruent”) Can be used after you have proven that two triangles are congruent. Can be used after you have proven that two triangles are congruent.

Example 1

Section 4-8: Introduction to Coordinate Proof Coordinate proof – a style of proof that uses coordinate geometry and algebra Coordinate proof – a style of proof that uses coordinate geometry and algebra

Example 1

Section 4-9 Isosceles and Equilateral Triangles Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent. Converse of Isosceles Theorem If two angles of a triangle are congruent, then the sides opposite the angles are congruent. angle B is congruent to angle C DE is congruent to DF

Example 1 2x° (x+30)° Find the angle measure. B. Measure of angle S M of angle S is congruent to M of angle R. 2x° = (x + 30)° x = 30 Thus M of angle S = 2x° = 2(30) = 60°. Isosceles Triangle Theorem Substitute the given values. Subtract x from both sides.