Congruence and Transformations

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Houghton Mifflin Harcourt Publishing Company

Holt McDougal Analytic Geometry

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Congruence and Transformations

TYPES OF TRANSFORMATIONS (centered at (0, 0))

Translation (slide): (x, y) (x a, y b) Reflection y-axis: (x, y) (x, y)

x-axis: (x, y) (x, y)

Rotation 90 clockwise: (x, y) (y, x) Dilation: (x, y) (kx, ky), k 0

Rotation 90 counterclockwise: (x, y) (y, x)

Rotation 180: (x, y) (x, y)

Apply the transformation M to the polygon with the given vertices.

Identify and describe the transformation.

1. M: (x, y) (x 1, y 2)

A(1, 3), B(2, 2), C(2, 1)

_____________________________________

2. M: (x, y) (x, y)

P(0, 0), Q(1, 3), R(3, 3)

_____________________________________

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Congruence and Transformations continued

An isometry is a transformation that preserves length, angle, and area. Because of these

properties, isometries produce congruent images. A rigid transformation is another name for

an isometry.

Dilations with scale factor k 1 are transformations that produce images that are not

congruent to their preimages.

Determine whether the polygons with the given vertices are

congruent.

3. E(3, 1), F(2, 4), G(0, 0)

H(1, 4), I(2, 1), J(4, 5)

___________________________________

4. R(2, 4), S(0, 3), T(3, 1)

U(2, 4), S(0, 3), V(3, 1)

_____________________________________

5. P(0, 0), Q(2, 2), R(2, 1)

P(0, 0), S(4, 4), T(4, 2)

_____________________________________

6. J(2, 2), K(2, 1), L(1, 3)

P(4, 4), Q(4, 2), R(2, 6)

__________________________________________

Transformation Isometry Image Preimage

translation yes yes

reflection yes yes

rotation yes yes

dilation no no

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Angle Relationships in Triangles

According to the Triangle Sum Theorem, the sum of the angle

measures of a triangle is 1808.

mJ mK mL 62 73 45

1808

The corollary below follows directly from the Triangle Sum Theorem.

Use the figure for Exercises 1 and 2.

1. Find mABC.

__________________________________________

2. Find mCAD.

__________________________________________

Use RST for Exercises 3 and 4.

3. What is the value of x?

__________________________________________

4. What is the measure of each angle?

__________________________________________

What is the measure of each angle?

5. L 6. C 7. W

________________________ ________________________ ________________________

Corollary Example

The acute angles of a right triangle are complementary.

mC mE 908

mC 90 39

518

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Angle Relationships in Triangles continued

An exterior angle of a triangle is formed by

one side of the triangle and the extension of

an adjacent side.

1 and 2 are the remote interior angles of

4 because they are not adjacent to 4.

Find each angle measure.

8. mG 9. mD

________________________________________ ________________________________________

Find each angle measure.

10. mM and mQ 11. mT and mR

________________________________________ ________________________________________

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.

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.

m4m1 1 m2

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Congruent Triangles Triangles are congruent if they have the same size and shape. Their corresponding parts,

the angles and sides that are in the same positions, are congruent.

To identify corresponding parts of congruent triangles, look at the order of the vertices in the

congruence statement such as ABC JKL.

Given: XYZ NPQ. Identify the congruent corresponding parts.

1. Z ______________________ 2. YZ ______________________

3. P ______________________ 4. X ______________________

5. NQ ______________________ 6. PN ______________________

Given: EFG RST. Find each value below.

7. x ______________________ 8. y ______________________

9. mF ______________________ 10. ST ______________________

Corresponding Parts

Congruent Angles Congruent Sides

A J

B K

C

L

AB JK

BC KL

CA LJ

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Congruent Triangles continued

You can prove triangles congruent by

using the definition of congruence.

Given: D and B are right angles.

DCE BCA

C is the midpoint of .DB

,ED AB EC AC

Prove: EDC ABC

Proof:

11. Complete the proof.

Given: Q R

P is the midpoint of .QR

,NQ SR NP SP

Prove: NPQ SPR

Statements Reasons

1. D and B are rt. s . 1. Given

2. D B 2. Rt.< Thm.

3. DCE BCA 3. Given

4. E A 4. Third s Thm.

5. C is the midpoint of DB . 5. Given

6. DC BC 6. Def. of mdpt.

7. ,ED AB EC AC 7. Given

8. EDC ABC 8. Def. of s

Statements Reasons

1. Q R 1. Given

2. NPQ SPR 2. a. ____________________________

3. N S 3. b. ____________________________

4. P is the midpoint of .QR 4. c. ____________________________

5. d. ____________________________ 5. Def. of mdpt.

6. ,NQ SR NP SP 6. e. ____________________________

7. NPQ SPR 7. f. ____________________________

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Triangle Congruence: SSS and SAS

You can use SSS to explain why FJH >

FGH.

It is given that FJ FG and that

.JH GH By the Reflex.

Prop. of , .FH FH So FJH FGH

by SSS.

Side-Side-Side (SSS) Congruence Postulate

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

, ,QR TU RP US and ,PQ ST so PQR STU.

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 another triangle, then the triangles are congruent.

Use SSS to explain why the triangles in each pair are congruent.

1. JKM LKM 2. ABC CDA

3.Use SAS to explain why WXY WZY.

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Show that the triangles are congruent for

the given value of the variable.

You can show that two triangles are congruent by using SSS and SAS

Show that JKL FGH for y 7

HG y6 mG 5y5 FG 4y 1

76 13 5(7)5 40 4(7) 1 27

HG LK 13, so HG LK by def. of segs. mG40,

so G K by def. of FG JK 27, so FG JK

by def. of segs. Therefore JKL FGH by SAS.

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Isosceles and Equilateral Triangles

You can use these theorems to find angle

measures in isosceles triangles.

Find mE in DEF.

mD mE Isosc. Thm.

5x (3x14) Substitute the

given values.

2x 14 Subtract 3x from

both sides.

x 7 Divide both

sides by 2.

Thus mE 3(7)14 35.

Find each angle

measure.

Theorem Examples

Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent.

If ,RT RS then

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Isosceles and Equilateral Triangles continued

Equilateral Triangle Corollary

If a triangle is equilateral, then it is equiangular.

(equilateral equiangular )

Equiangular Triangle Corollary

If a triangle is equiangular, then it is equilateral.

(equiangular equilateral )

If

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Properties of Parallelograms

A parallelogram is a quadrilateral with two pairs

of parallel sides.

All parallelograms, such as FGHJ, have

the following properties.

Properties of Parallelograms

FG HJ

GH JF

Opposite sides are congruent.

F H

G J Opposite angles are congruent.

mF mG 180

mG mH 180

mH mJ 180

mJ mF 180 Consecutive angles are supplementary.

FP HP

GP JP

The diagonals bisect each other.

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