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Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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)
_____________________________________
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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
Name _______________________________________ Date ___________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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. ____________________________
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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.
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
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.
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Reteach
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.
Name _______________________________________ Date __________________ Class __________________
Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
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