Chapter 4Chapter 4Triangle Triangle

CongruenceCongruence

By: Maya RichardsBy: Maya Richards55thth Period Geometry Period Geometry

Section 4-1: Congruence and Section 4-1: Congruence and TransformationsTransformations

Transformations:Transformations: Translations – slidesTranslations – slides Reflections – flipsReflections – flips Rotations – turnsRotations – turns Dilations – gets bigger or smaller (only one that changes Dilations – gets bigger or smaller (only one that changes

size)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 TrianglesSection 4-2: Classifying Triangles

Example 1Example 1

30°

30°

60° 60°

Classify each triangle by its angle measures.

A. Triangle EHGAngle EHG is a right angle, so triangle EHG is a right triangle.

B. Triangle EFHAngle 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 2Example 2

15

15 5

18

Classify each triangle by its side lengths.

A. Triangle ABCFrom the figure, AB is congruent to AC.So AC = 15, and triangle ABC is equilateral.

B. Triangle ABDBy the Segment Addition Postulate, BD = BC + CD = 15 + 5 = 20.Since no sides are congruent, triangle ABD is scalene.

Section 4-3: Angle Relationships in Section 4-3: Angle Relationships in TrianglesTriangles

Triangle Sum TheoremThe 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 TheoremIf 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 1Example 1

Section 4-4: Congruent TrianglesSection 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: Section 4-5: Triangle Congruence: SSS and SASSSS and SAS

Side-Side-Side Congruence (SSS)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 1Example 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: Section 4-6: Triangle Congruence: ASA, AAS, and HLASA, 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 1Example 1

Section 4-7: Triangle Congruence: Section 4-7: Triangle Congruence: CPCTCCPCTC

CPCTC = (“Corresponding Parts of Congruent CPCTC = (“Corresponding Parts of Congruent Triangles are Congruent”)Triangles are Congruent”)

Can be used after you have proven that two Can be used after you have proven that two triangles are congruent.triangles are congruent.

Example 1Example 1

Section 4-8: Introduction to Section 4-8: Introduction to Coordinate ProofCoordinate Proof

Coordinate proof – a style of proof that uses Coordinate proof – a style of proof that uses coordinate geometry and algebracoordinate geometry and algebra

Example 1Example 1

Section 4-9 Isosceles and Equilateral Section 4-9 Isosceles and Equilateral TrianglesTriangles

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 1Example 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 = 30Thus M of angle S = 2x° = 2(30) = 60°.

Isosceles Triangle Theorem

Substitute the given values.

Subtract x from both sides.