1. 4.1 Quadrilaterals Quadrilateral Parallelogram Trapezoid
Isosceles Trapezoid
Rhombus
Rectangle
Square

2. 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

3. 4.1 Properties of a Parallelogram
Opposite angles are congruent
Opposite sides are congruent
Diagonals bisect each other
Consecutive angles are supplementary

4. 4.1 Properties of a Parallelogram
In the following parallelogram: AB = 7, BC = 4,
What is CD?
What is AD?
What is mABC?
What is mDCB? A B D C

5. 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

6. 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.

7. 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 C B

8. 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).

9. 4.3 Rectangle, Square, and Rhombus
A rhombus is a parallelogram with all sides congruent.
The diagonals of a rhombus are perpendicular.

10. 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 + b2 Note: You can use this to get the length of the diagonal of a rectangle. a c b

11. 4.4 The Trapezoid
Definition: A trapezoid is a quadrilateral with exactly 2 parallel sides.
Base
Leg
Leg
Base
Base angles

12. 4.4 The Trapezoid Isosceles trapezoid: 2 legs are congruent
Base angles are congruent
Diagonals are congruent

13. 4.4 The Trapezoid A B
Median of a trapezoid: connecting midpoints of both legs M N D C

14. 4.4 Miscellaneous Theorems
If 3 or more parallel lines intercept congruent segments on one transversal, then they intercept congruent segments on any transversal.

15. 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 quantities example: sides of a triangle are in the ratio 2:3:4

16. 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 extremes where a,d are the extremes and b,c are the means (a.k.a. “cross-multiplying”)

17. 5.1 Ratios, Rates, and Proportions
b is the geometric mean of a & c
…..used with similar triangles

18. 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

19. 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”

20. 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

21. 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)

22. 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

23. 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.

24. 5.3 Proving Triangles Similar
(example proof using CSSTP)
Statements
Reasons
1. mA = m D
Given
2. mB = m E
2. Given
3. ABC ~ DEF
3. AA 4.
4. CSSTP

25. 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

26. 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

27. 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 + b2  the triangle is obtuse
c2 < a2 + b2  the triangle is acute
c2 = a2 + b2  the triangle is right c a b

28. 5.5 Special Right Triangles
Leg opposite the 45 angle = a
Leg opposite the 90 angle =
45 a
90
45 a

29. 5.5 Special Right Triangles
Leg opposite 30 angle = a
Leg opposite 60 angle =
Leg opposite 90 angle = 2a
60 2a a
30
90

30. 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

31. 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

32. 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