Section 4.1 Congruent Polygons. Polygons Examples of Polygons Polygons Examples of Non-Polygons Non-Polygons.

Section 4.1

Congruent Polygons

Polygons

Examples of PolygonsPolygons

Examples of Non-PolygonsNon-Polygons

Naming a Polygon

Octagon CDEFGHAB H A

G B

F C

E D

When naming polygons, the rule is to go around the figure, either clockwise or counterclockwise, and list the vertices in order. It does not matter which vertex you list first.

Corresponding Sides and Angles• If two polygons have the same number of

sides, it is possible to set up a correspondence between them by pairing their parts. In rhombi RSTU and MNOP, the corresponding angles and sides would be as follows.

Pair – Angles R and M, S and N, T and O, and U and PR M

U S P N

T O

Polygon Congruence Postulate

Two Polygons are congruent if and only if there is a correspondence between their sides and angles such that:

Each pair of corresponding angles is congruent.Each pair of corresponding sides is congruent.

Otherwise, same shape, same size.

Congruence Statement

• ABCDE KRMNG, Angle A corresponds to K, B ≅to R, C to M, D to N, and E to G.

A N

E B M G

D C R K

Section 4.2

Triangle Congruence

SSS (Side-Side-Side) Postulate

• If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent. VEG TFH△ ≅△

G H 12m 12m

19m V 19m T

20m 20m E F

SAS (Side-Angle-Side) Postulate

• If two sides and their angle in one triangle are congruent to two sides and their angle in another triangle, then the two triangles are congruent. ABC FDE△ ≅△

A F 37˚ 37˚

C E B D

ASA (Angle-Side-Angle) Postulate

• If two angles and the side between the two angles in one triangle are congruent to two angles and the side between the two angles in another triangle, then the two triangles are congruent. MNO TSR△ ≅△

M S R 56° 39° O 39° 56° N T

Section 4.3

Analyzing Triangle Congruence

AAS (Angle-Angle-Side) Postulate

• If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent. ABC EFD △ ≅△

A F 101° 35° D

C 35° 101° B E

Combinations That Do Not Work Unless Under Special Circumstances

• AAA (Angle-Angle-Angle)• SSA (Side-Side-Angle)

When Dealing With Right Triangles

• HL (Hypotenuse-Leg) Congruence Theorem

• If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are congruent.

• Other Possibilities

• LL (Leg-Leg) Congruence Theorem

• LA (Leg-Angle) Congruence Theorem

• HA (Hypotenuse-Angle) Congruence Theorem

Section 4.4

Using Triangle Congruence

CPCTC in Flowchart Proofs

• CPCTC – Corresponding Parts of Congruent Triangles are Congruent.

• CPCTC is used after a triangle congruence postulate (SSS, ASA, SAS, or AAS) has been establish to prove two triangles are congruent.

• Flowchart Proofs are used to explain and understand why two or more triangles are congruent.

Given: AC BD, CX DX, and <C <D≅ ≅ ≅Prove: X is the midpoint of AB.

C

B X

A

D

X is the midpoint of AB by Def. of midpoint

AX BX by CPCTC≅△ACX≅△BDX by SASAC≅BD GivenCX≅DX Given<C≅<D Given

Isosceles Triangle

• An isosceles triangle is a triangle with two congruent sides.

• The two congruent sides are known as the legs of the triangle.

• The remaining side is known as the base.• The angle opposite the base is the vertex angle.• The angles whose vertices are the endpoints of

the base are base angles.

Theorems Involving Isosceles Triangles

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

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

• The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

Reminders For an Equilateral Triangle

• The measure of each angle of an equilateral triangle is 60°.

• The measures of the sides are congruent.

Section 4.5

Proving Quadrilateral Properties

Quadrilateral Hierarchy

Quadrilateral – polygon with 4 sides

Quad’s with no pairs of parallel sides

Kites

Trapezoids - one pair of parallel sides

Isosceles Trapezoids

Parallelograms - two pair of parallel sides

Rectangles

Squares

Rhombi

Properties of Parallelograms

• A diagonal divides the parallelogram into two congruent triangles.

• Opposite sides of a parallelogram are ≅.

• Opposite angles of a parallelogram are ≅.

• Consecutive angles are supplementary.

• The two diagonals bisect each other.

• Rectangles, rhombi, and squares are parallelograms.

Properties of Parallelograms

Properties of Rectangles• The diagonals are ≅.• Four 90° angles.

Properties of Rhombi• The diagonals are

perpendicular.• Four ≅ sides.• The diagonals bisect a pair

of opposite angles.

Properties of Squares

• All properties of a parallelogram• All properties of a rectangle• All properties of a rhombus

Section 4.6

Conditions for Special Quadirlaterals

The Conditions That Determine a Figure

• If you are given a quadrilateral, check the given information against the definitions of the special figures.

• Check for properties that match or you can prove to a special figure.

• It is better to be more specific than general. Remember a square is always a parallelogram but a parallelogram is not always a square.

The Triangle Midsegment Theorem

• A midsegment of a triangle is a parallel to a side of the triangle and has a measure equal to half of the measure of that side.

• CD is the midsegment of AGH.△ G H

C D

A

Review of Other Properties

• Substitution Property• Transitive Property• Reflexive Property• Symmetric Property• Transitive Property• SSS (Side-Side-Side)• ASA (Angle-Side-Angle)• SAS (Side-Angle-Side)

• AAS (Angle-Angle-Side)• CPCTC • Definition of Midpoint• Alt. Interior Angles• Cons. Interior Angles• Corresponding Angles• Alt. Exterior Angles• Vertical Angles Theorem

Section 4.7

Compass and Straightedge Constructions

Compass and Straightedge Constructions

• A segment congruent to a given segment pg. 261

• A triangle congruent to a given triangle pg. 262

• Angle bisector pg. 263• An angle congruent to a

given angle pg. 265

• The perpendicular bisector of a given segment and the midpoint of a given segment pg. 266

• A line through a point perpendicular to a given line pg. 267

• A line through a point parallel to a given line pg. 268

Section 4.8

Constructing Transformations

Transformations

• Rotation (Turn) – every point of the preimage is rotated by a given angle about a point.

• Reflection (Flip) – every point of the preimage may be connected to its image point by a segment that (a) is perpendicular to the line or plane that is the mirror of the reflection and (b) has its midpoint on the mirror of the reflection.

Transformations

• Translation (Slide) – every point of the preimage moves in the same direction by the same amount to form the image.

• Dilation – a figures size is increased or decreased by a scale factor, but its shape stays the same.

Other Theorems

Betweenness Postulate • Converse of the Segment

Addition Postulate• Given three points P, Q, and

R, if PQ + QR = PR, then P, Q, and R are collinear and Q is between P and R.

Triangle Inequality Theorem• The sum of the lengths of

any two sides of a triangle is greater than the length of the third side.

• AB + BC > AC A• AB + AC > BC• AC + BC > AB B

C