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Section 4-1: Congruence and Transformations

Transformations:

Translations – slides

Reflections – flips

Rotations – turns

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.

Reflection across the y-axis.

Dilation of 2x.

Example 1

H

Classify each triangle by its angle measures.

30°

Triangle EHG

Angle EHG is a right angle, so triangle EHG is a right triangle.

120°

E

30°

60°

60°

G

F

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.

Example 2

Classify each triangle by its side lengths.

A

18

15

D

B

5

C

15

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 = 15 + 5 = 20.

Since no sides are congruent, triangle ABD is scalene.

Section 4-3: Angle Relationships in Triangles

B

Triangle Sum Theorem

The sum of the angle measures of a triangle is 180 degrees.

- angle A + angle B + angle C = 180°

A

C

- 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.

2

1

3

4

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

N

M

R

T

S

Section 4-4: Congruent Triangles

C

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

A

E

B

D

F

Section 4-5: Triangle Congruence: SSS and SAS

A

Side-Side-Side Congruence (SSS)

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

B

C

E

F

D

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.

B

A

C

F

D

E

Example 1

Q

Use SSS to explain why triangle PQR is congruent to triangle PSR.

P

R

S

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.

B

C

A

D

F

E

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.

B

A

F

C

D

E

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.

B

A

F

C

D

E

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.

Section 4-8: Introduction to Coordinate Proof

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

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.

A

angle B is congruent to angle C

B

C

D

DE is congruent to DF

E

F

Example 1

(x+30)°

R

Find the angle measure.

T

2x°

S

B. Measure of angle S

M of angle S is congruent to M of angle R.

Isosceles Triangle Theorem

2x° = (x + 30)°

Substitute the given values.

x = 30

Subtract x from both sides.

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

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