Secondary Math II – Honors

Unit 4 Notes

Polygons

Name:____________________

Per:_____

Unit 4 Notes / Secondary 2 Honors

Day 1: Interior and Exterior Angles of a Polygon Vocabulary: Polygon: ____________________________________________________________________________________ Regular Polygon:______________________________________________________________________________ Example(s): Discover the formula for the sum of interior angles of a polygon. 1. Determine the sum of the interior angles of a pentagon by completing each step. a. Draw a pentagon. Then draw all possible diagonals using only one vertex of the polygon. b. How many triangles are formed? ______

c. The sum of the interior angles of ONE triangle = ______. Since a pentagon has ____ triangles, the sum of all interior angles in a pentagon is ___________________.

2. Determine the sum of the interior angles of a hexagon by completing each step. a. Draw a hexagon. Then draw all possible diagonals using only one vertex of the polygon. b. How many triangles are formed? ______ c. The sum of the interior angles of ONE triangle = ______. Since a hexagon has ____ triangles, the sum of all interior angles in a hexagon is __________________.

Polygon # of Sides Number of Triangles Sum of interior angles

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

n-gon

Find the sum of the measures of the interior angles of each of the following. 3. 14-gon 4. 25-gon 5. 32-gon Find the value of x in the figures below. 6. 7. The sum of the measures of the interior angles of a polygon is given. Determine the number of sides for each polygon.

8. 1620 9. 2700 10. 1080 For each regular polygon, calculate the measure of each of its interior angles. 11. 12.

Vocabulary: Exterior angle of a polygon: Formed adjacent to each interior angle by extending one side of each vertex of the polygon. Interior Angle + Exterior Angle = ______ Example(s): Given each interior angle, calculate the measure of the adjacent exterior angle.

13. interior angle is 100 14. interior angle measures 80

Discover the sum of exterior angles of any polygon: 15. Given the triangle to the right answer the following: a. How many linear pairs are formed? b. What is the sum of the linear pairs? c. What is the sum of all the interior angles? d. What is the sum of the exterior angles? 16. Draw a quadrilateral and extend each side to locate an exterior angle at each vertex (like the picture above). a. How many linear pairs are formed? b. What is the sum of the linear pairs? c. What is the sum of all the interior angles? d. What is the sum of the exterior angles? 17. Complete the table.

Number of Sides of the Polygon 3 4 5 6 15

Number of Linear Pairs Formed

Sum of Measures of Linear Pairs

Sum of Measures of Interior Angles

Sum of Measures of Exterior Angles

Find the measure of each exterior angle of the regular polygon. 18. 19. 20. If a regular polygon has 100 sides, calculate the measure of each exterior angle.

Calculate the number of sides for each regular polygon given the following information.

21. The measure of each exterior angle is 30 22. The measure of each exterior angle is18 . 23. The measure of each interior angle is 156°. 24. The measure of each interior angle is 162°. Summary: **To find the sum of interior angles use _______________________. **The sum of exterior angles is always _________. **If you have a regular polygon then _______ divided by ____ will give you the measure of each exterior angle. Day 2: Squares, Rectangles, Parallelograms, & Rhombi Vocabulary: Quadrilateral: _____ sided polygon. Parallelogram: A quadrilateral with ______________ pairs of ________________sides __________________. Rectangle: A quadrilateral with opposite sides ___________ and ____________ and all angles are ______________. Square: A quadrilateral with _____ _____________ _____________ and all sides congruent. Rhombus: A quadrilateral with _______ _______________sides.

Properties of a Parallelogram:

Opposite sides are congruent

_______ ≅ _______ _______ ≅ _______

Opposite sides are parallel _______ ∥_______ _______ ∥_______

Opposite angles are congruent _______ ≅ _______ _______ ≅ _______

Diagonals bisect each other _______ ≅ _______ _______ ≅ _______

Consecutive angles are supplementary _____+ ____= 180° _____+ ____= 180 _____+ ____= 180° _____+ ____= 180°

Find the missing measures using the parallelogram below. 1) _____x

2) _____h

3) _____m B

Find the missing measures using the parallelogram below. 4) x = ______ 5) a = _____ 6) f = ______ 7) BD = ______

Properties of a Rectangle:

Opposite sides are congruent AND parallel _______ ≅ _______ _______ ≅ _______ _______ ∥________ _______ ∥________

All angles congruent (90 degrees) _______ ≅ _______ ≅_______ ≅ ______

Consecutive angles are supplementary _____+ _____= 180° _____+ _____= 180° _____+ _____= 180° _____+ _____= 180°

Diagonals bisect each other _______ ≅ _______ _______ ≅ _______

Given ABCD is a rectangle determine the following: 8) ∠𝐴𝐷𝐶 = _______ 9) DC= ______ 10) BD = ______ 11) ∠DEA = _______

Properties of a Square:

All sides are congruent _______ ≅ _______ ≅_______ ≅ ______

All angles are congruent _______ ≅ _______ ≅_______ ≅ ______

Opposite sides are parallel _______ ∥_______ _______ ∥_______

Consecutive angles are supplementary _____+ _____= 180° _____+ _____= 180° _____+ _____= 180° _____+ _____= 180°

The diagonals are congruent _______ ≅ _______

The diagonals bisect each other _______ ≅ _______ _______ ≅ _______

The diagonals bisect the vertex angles _______ ≅ _______ ≅_______ ≅ ______ ≅_______ ≅ ______ ≅_______ ≅ ______

The diagonals are perpendicular to each other _______ ⊥ _______

Properties of a Rhombus:

Opposite angles are congruent _______ ≅ _______ _______ ≅ _______

Opposite sides are parallel _______ ∥_______ _______ ∥_______

All sides are congruent _______ ≅ _______ ≅_______ ≅ ______

The diagonals bisect the vertex angles _______ ≅ _______ _______ ≅ ______ _______ ≅ ______ _______ ≅ ______

The diagonals bisect each other _______ ≅ _______ _______ ≅ _______

The diagonals are perpendicular to each other

_______ ⊥ _______

Use rhombus SPQR to determine the following: 12) SP = ______ 13) If m∠QPS = ______ 14) Complete a two-column proof to prove that 𝐴𝐶̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅ in square ABCD ** Summary: Replace the question marks with the correct vocabulary term. Use rectangle STUV to answer the following: 15) If m∠1 = 30, m∠2 = _______ 16) If m∠6 = 57, m∠4 = _______ 17) If m∠8 = 133, m∠2 = _______ 18) If m∠5 = 16, m∠3 = _______

1( )

2 EF BC AD

19) ABCD is a rhombus. If the perimeter of ABCD = 68 in. and BD = 16in, find AC. 20) ABCD is a square. If m∠DBC = x - 4, find x. Day 3: Properties of Kites and Trapezoids Vocabulary:

• Kite: A quadrilateral with ________ pairs of consecutive ______________ sides, but ___________________ sides ARE NOT congruent.

• Trapezoid: A quadrilateral with exactly _____ pair of _________________ sides. The parallel sides are called the ________________. The nonparallel sides are called the _______________.

• Isosceles Trapezoid: A trapezoid with congruent __________________ sides.

• Midsegment of a Trapezoid is parallel to each base and its length is _______________________________

Properties of a Kite:

One pair of opposite angles of a kite is congruent _____≅ _____

The diagonals are perpendicular to each other ______ ⊥ ______

The diagonal that connects the opposite vertex angles that are not congruent bisects the diagonal that connects the opposite vertex angles that are congruent. ______≅ ______

The diagonal that connects the opposite vertex angles that are not congruent bisects the vertex angles. ______≅ ______

______≅ ______

Example(s): 1. Given ABCD is a kite, determine the length of segments CD, AD, CB, AB, and DB, and AC. 2. Find the perimeter of the kite. Round answer to the nearest inch. Given trapezoid ABCD , find the other angles: 3. 4. 5. Maria told Sam that an isosceles trapezoid must also be a parallelogram because there is a pair of congruent sides in an isosceles trapezoid. Is Maria correct? Explain? Given trapezoid ABCD find x. 6. 7. 8. Write a two-column proof to prove 𝐻𝐹̅̅ ̅̅ ≅ 𝐽𝐹̅̅ ̅ in kite GHIJ.

Statements Reasons

1. Kite GHIJ with diagonals 𝐻𝐽 ̅̅ ̅̅ and 𝐺𝐼̅̅ ̅ intersecting at point F.

1.

2. 2. Definition of a kite

3. 3.

4. ∆𝐺𝐻𝐼 ≅ ∆𝐺𝐽𝐼 4.

5. 5. CPCTC

6. 𝐺𝐹̅̅ ̅̅ ≅ 𝐺𝐹̅̅ ̅̅ 6.

7. 7. SAS

8. 8.

9. Complete the flow chart for quadrilaterals. 10. Place a check mark in the box that has the given characteristic.

Characteristic Trapezoid Parallelogram Kite Rhombus Rectangle Square

No parallel sides

Exactly one pair of parallel sides

Two pairs of parallel sides

One pair of sides are both congruent and parallel

Two pairs of opposite sides are congruent

Exactly one pair of opposite angles are congruent

Two pairs of opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other

All sides are congruent

Diagonals are perpendicular to each other

Diagonals bisect the vertex angles

All angles are congruent

Diagonals are congruent

Determine whether each statement is true or false. If false, explain why. 11. A square is also a rectangle. 12. The base angles of a trapezoid are always congruent. 13. The diagonals of a rhombus bisect each other. 14. A parallelogram is also a trapezoid. 15. Classify the quadrilaterals by their properties. State its most specific name.