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4.1 Quadrilaterals

Quadrilateral

Parallelogram

Trapezoid

IsoscelesTrapezoid

Rhombus

Rectangle

Square

4.1 Properties of a Parallelogram

- Definition: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

A

B

D

C

4.1 Properties of a Parallelogram

- Properties of a parallelogram:
- Opposite angles are congruent
- Opposite sides are congruent
- Diagonals bisect each other
- Consecutive angles are supplementary

4.1 Properties of a Parallelogram

- In the following parallelogram:AB = 7, BC = 4,
- What is CD?
- What is AD?
- What is mABC?
- What is mDCB?

A

B

D

C

4.2 Proofs

- Proving a quadrilateral is a parallelogram:
- Show both pairs of opposite sides are parallel (definition)
- Show one pair of opposite sides are congruent and parallel
- Show both pairs of opposite sides are congruent
- Show the diagonals bisect each other

4.2 Kites

- Kite - a quadrilateral with two distinct pairs of congruent adjacent sides.
- Theorem: In a kite, one pair of opposite angles is congruent.

4.2 Midpoint Segments

- The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to ½ the length of the third side.

A

M

N

B

C

4.3 Rectangle, Square, and Rhombus

- Rectangle - a parallelogram that has 4 right angles.
- The diagonals of a rectangle are congruent.
- A square is a rectangle that has all sides congruent (regular quadrilateral).

4.3 Rectangle, Square, and Rhombus

- A rhombus is a parallelogram with all sides congruent.
- The diagonals of a rhombus are perpendicular.

4.3 Rectangles: Pythagorean Theorem

- Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2Note: You can use this to get the length of the diagonal of a rectangle.

a

c

b

4.4 The Trapezoid

- Definition: A trapezoid is a quadrilateral with exactly 2 parallel sides.

Base

Leg

Leg

Base

Base angles

4.4 The Trapezoid

- Isosceles trapezoid:
- 2 legs are congruent
- Base angles are congruent
- Diagonals are congruent

4.4 Miscellaneous Theorems

- If 3 or more parallel lines intercept congruent segments on one transversal, then they intercept congruent segments on any transversal.

5.1 Ratios, Rates, and Proportions

- Ratio - sometimes written as a:b Note: a and b should have the same units of measure.
- Rate - like ratio except the units are different(example: 50 miles per hour)
- Extended Ratio: Compares more than 2 quantitiesexample: sides of a triangle are in the ratio 2:3:4

5.1 Ratios, Rates, and Proportions

- two rates or ratios are equal (read “a is to b as c is to d”)
- Means-extremes property:product of the means = product of the extremeswhere a,d are the extremes and b,c are the means(a.k.a. “cross-multiplying”)

5.1 Ratios, Rates, and Proportions

- b is the geometric mean of a & c
- …..used with similar triangles

5.1 Ratios, Rates, and Proportions

- Ratios – property 2:(means and extremes may be switched)
- Ratios – property 3:Note: cross-multiplying will always work, these may lead to a solution faster sometimes

5.2 Similar Polygons

- Definition: Two Polygons are similar two conditions are satisfied:
- All corresponding pairs of angles are congruent.
- All corresponding pairs of sides are proportional.Note: “~” is read “is similar to”

5.2 Similar Polygons

- Given ABC ~ DEF with the following measures, find the lengths DF and EF:

E

10

5

B

D

A

6

4

C

F

5.3 Proving Triangles Similar

- Postulate 15: If 3 angles of a triangle are congruent to 3 angles of another triangle, then the triangles are similar (AAA)
- Corollary: If 2 angles of a triangle are congruent to 2 angles of another triangle, then the triangle, then the triangles are similar. (AA)

5.3 Proving Triangles Similar

- AA - If 2 angles of a triangle are congruent to 2 angles of another triangle, then the triangle, then the triangles are similar.
- SAS~ - If a an angle of one triangle is congruent to an angle of a second triangle and the pairs of sides including the two angles are proportional, then the triangles are similar

5.3 Proving Triangles Similar

- SSS~ - If the 3 sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar
- CSSTP – Corresponding Sides of Similar Triangles are Proportional (analogous to CPCTC in triangle congruence proofs)
- CASTC – Corresponding angles of similar triangles are congruent.

5.3 Proving Triangles Similar

- (example proof using CSSTP)

5.4 Pythagorean Theorem

- Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2
- Converse of Pythagorean Theorem: If for a triangle, c2 = a2 + b2 then the opposite side c is a right angle and the triangle is a right triangle.

c

a

b

5.4 Pythagorean Theorem

- Pythagorean Triples: 3 integers that satisfy the Pythagorean theorem
- 3, 4, 5 (6, 8, 10; 9, 12, 15; etc.)
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25

5.4 Classifying a Triangle by Angle

- If a, b, and c are lengths of sides of a triangle with c being the longest,
- c2 > a2 + b2the triangle is obtuse
- c2 < a2 + b2the triangle is acute
- c2 = a2 + b2the triangle is right

c

a

b

5.5 Special Right Triangles

- 45-45-90 triangle:
- Leg opposite the 45 angle = a
- Leg opposite the 90 angle =

45

a

90

45

a

5.5 Special Right Triangles

- 30-60-90 triangle:
- Leg opposite 30 angle = a
- Leg opposite 60 angle =
- Leg opposite 90 angle = 2a

60

2a

a

30

90

5.6 Segments Divided Proportionally

- If a line is parallel to one side of a triangle and intersects the other two sides, then it divides these sides proportionally

A

D

E

B

C

5.6 Segments Divided Proportionally

- When 3 or more parallel lines are cut by a pair of transversals, the transversals are divided proportionally by the parallel lines

A

D

B

E

C

F

5.6 Segments Divided Proportionally

- Angle Bisector Theorem: If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the length of the 2 sides which form that angle.

C

A

B

D

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