Angles in Polygons

Day

/ In Class / Homework / Completed
Day 1 / Angles in Polygons / SP 6-1 (1-3, 9-16)
Day 2 / Angles in Polygons / SP 6-1 (5-8)
Day 3 / Parallelograms / D3 HW – Pg. 9
Day 4 / Parallelograms / SP 6-3 (1-4, 8-11)
Day 5 / Parallelograms / D5 HW – Pg. 14
Day 6 / Rectangles / SP 6- 4 #1-8
Day 7 / Rectangles / D7 HW – Pg. 18
Day 8 / Rhombi and Squares / SP 6- 5 #1-6
Day 9 / Rhombi and Squares / D9 HW – Pg. 25 #1
Day 10 / Trapezoids / D10 HW – Pg. 29 #2
Day 11 / Trapezoids / SP 6-6 #1-8
Day 12 / Review
Day 13 / Test /

Good Luck!

/

Angles of Polygons

A diagonal of a polygon is a segment that connects any two nonconsecutive vertices.

The sum of the interior angles of a triangle is ______.

Recall the Triangle-Sum Theorem:

The sum of the angle measures of a triangle is ______.

1.  In triangle ABC, the angles are in the ratio of 4:5:11. Find the measure of each angle and classify the triangle according to its angles.

The sum of the angle measures of a polygon is the sum of the angle measures of the triangles formed by drawing all the possible diagonals from one vertex.

1 triangle 2 triangles 3 triangles 4 triangles

1.  Find the sum of the measures of the interior angles of a pentagon.

2.  Find the sum of the measures of the interior angles of a hexagon.

3.  Find the value of x.

4.  Find the measure of each interior angle of quadrilateral ABCD.

5.  Find the value of x.

Recall: Regular polygon means all sides ______and all angles ______.

1.  Find the measure of each interior angle of a regular heptagon.

2.  Find the measure of each interior angle of a regular 12-gon.

An interior and an exterior angle form a linear pair. They are ______angles. They add up to _____.

What is the sum of the exterior angles in the following diagrams?

1.  Find the value of x.

1.  Find the measure of each exterior angle of a regular nonagon.

Find the Number of Sides Given Interior Angle Measure

1.  The measure of an interior angle of a regular polygon is 135. Find the number of sides in a polygon.

2.  The measure of an interior angle of a regular polygon is 144. Find the number of sides in a polygon.

3.  The measure of an interior angle of a regular polygon is 156. Find the number of sides in a polygon.

Regular Polygons

Polygon

/ # of Sides (n) / Sum of the Interior Angle Measure
180(n – 2) / One Interior Angle Measure
180(n – 2)/n / One Exterior Angle Measure
360/n
Triangle / 180(3 – 2) = 180
Quadrilateral
5
Hexagon
7
8
Nonagon
Decagon
12-gon
n-gon / n

Quadrilateral Family Tree

Parallelograms

A parallelogram is a quadrilateral with two pairs of parallel sides.

Parallel lines have the ______slope.

AB _____ DC and AD _____ BC

1.  Find the degree of the measure of the other three angles of a parallelogram if one angle measures 68.

2.  Given parallelogram ABCD, mÐB = x + 20 and mÐC = 3x. Find all the angle measures?

3.  In parallelogram ABCD, mÐA = 4x – 17 and mÐC = 2x – 5. Find the value of x.

4.  Find x and y so that the quadrilateral is a parallelogram.

a. b.

5.  In parallelogram ABCD, AB = 7x – 4 and CD = 2x + 21. Find AB and CD.

6.  In parallelogram ABCD, AC and BD intersect at E. If AE = 5x – 3 and EC = 15 – x, find AC.

7.  In parallelogram FGHJ, diagonals FH and JG intersect at K. If FK = 3x – 1, KG = 4y + 3, JK = 6y – 2, and KH = 2x + 3, find x and y.

8.  Find x and y.

9.  In the accompanying diagram of parallelogram MATH, mÐT = 100 and SH bisects ÐMHT. What is mÐHSA?

10. In the accompanying diagram of parallelogram ABCD, AB is extended to E such that CB @ CE. If mÐADC = 100, what is mÐCEB?

11. In parallelogram LMNO, an exterior angle at vertex O measures 72. Find the measure of ÐL.

Day 3 HW

1.  Find all the angle measures.

2.  In parallelogram TVWS, TV = 3x – 17 and SW = x – 11. Find the value of x.

3.  In parallelogram ABCD, mÐB = 4x + 10 and mÐD = 2x + 30. Find mÐB.

4.  In parallelogram KLMN, mÐL = x - 5 and mÐM = 2x + 11. Find mÐM.

5.  In parallelogram ABCD, AC and BD intersect at E. If AE = x + 9 and EC = 2x + 7, find AC.

6.  Factor: 4x2 – 36y2

7.  Find the sum of the measures of the interior angles of a 20-gon.

To Do Coordinate Geometry Proofs:

1.  Graph the figure.

2.  Use one or a combination of the following formulas:

Ø  Distance formula to show that ______are ______.

Ø  Slope formula to show that ______are ______or have ______.

Ø  Midpoint formula to show that ______have the ______midpoint.

3.  Show ALL work.

4. Write a statement(s) to explain why it is that figure to finish the proof.

Parallelograms

Definition/Property / Picture / Show What? / Formula(s) and Statement(s) Used?
If a quadrilateral is a parallelogram, then opposite sides are _____. / Statement:
If a quadrilateral is a parallelogram, then opposite sides are _____. / Statement:
If a quadrilateral is a parallelogram, then diagonals ______each other. / Statement:
If a quadrilateral is a parallelogram, then opposite angles are _____.
If a quadrilateral is a parallelogram, then consecutive angles between parallel lines are ______.

**********Choose only ONE method

1.  Graph quadrilateral KLMN with vertices K(2, 3), L(8, 4), M(7, -2), N(1, -3). Determine whether the quadrilateral is a parallelogram. Justify your answer using the slope formula.

2.  Graph quadrilateral ABCD with vertices A(3, 3), B(8, 2), C(6, -1), D(1, 0). Determine whether the quadrilateral is a parallelogram. Justify your answer using the distance formula.

3.  Quadrilateral ABCD has vertices A(1, 2), B(2, 5), C(5, 7), and D(4, 4). Prove that ABCD is a parallelogram by showing that the opposite sides are parallel.

4.  Quadrilateral ABCD has vertices A(1, 2), B(2, 5), C(5, 7), and D(4, 4). Prove that ABCD is a parallelogram by showing that the opposite sides are congruent.

5.  Quadrilateral ABCD has vertices A(1, 2), B(2, 5), C(5, 7), and D(4, 4). Prove that ABCD is a parallelogram by showing that the diagonals bisect each other.

6.  Prove that quadrilateral ABCD is a parallelogram using any method. A(–1, 1), B(–3, 4), C(1, 5), and D(3, 2).

Day 5 HW

1.  Quadrilateral ABCD has vertices A(–1, 7), B(8, 7), C(6, -2), and D(-3, -2). Prove that quadrilateral ABCD is a parallelogram. Show opposite sides are congruent.

2.  Quadrilateral ABCD has vertices A(1, 1), B(2, 3), C(6, 3), and D(5, 1). Prove that quadrilateral ABCD is a parallelogram. Show opposite sides are parallel.

1.  In rectangle RECT, the mÐREC=28 + 2a. Find the value of a.

2.  In rectangle ABCD, CB = 6, AB = 8, and AC = 10. Find:

a. AD b. CD c. EC d. AE

e. DE f. EB g. DB

3.  In rectangle ABCD, diagonals AC and BD intersect at point E. If AE = 20 and BD = 2x + 30, find x.

4.  In rectangle ABCD, AC = 2x + 15 and BD = 4x – 5. Find x.

5.  The lengths of the diagonals of a rectangle are represented by 2x + 3 and 4x – 11. Find the length of the diagonals.

6.  If AE = 3y – 5 and DB = 5y + 1, find y.

7.  Quadrilateral ABCD is a rectangle. If mÐCAB = 2x + 4 and mÐACB = 7x + 5, find x.

8.  Quadrilateral PQRS has vertices P(-5, 3), Q(1, -1), R(-1, -4), and S(-7, 0). Prove that quadrilateral PQRS is a rectangle. Hint: First prove it is a parallelogram.

.

9.  Quadrilateral JKLM has vertices J(-10, 2), K(-8, -6), L(4, -3), and M(2, 5). Prove that quadrilateral JKLM is a rectangle.

D7 HW

1.  Quadrilateral ABCD has vertices A(–3, 0), B(-2, 3), C(4, 1), and D(3, -2). Prove that quadrilateral ABCD is a rectangle.

2.  Quadrilateral ABCD has vertices A(-1, 0), B(0, 2), C(4, 0), and D(3, -2). Prove that quadrilateral ABCD is a rectangle.

1.  Given rhombus ABCD, find each length and each angle.

AB= mÐ2=

BC= mÐ4=

CD= mÐ3=

AD= mÐDAB=

mÐABC= mÐDEC=

2.  In rhombus ABCD, mÐBCD = 64. Find mÐBAC.

3.  In rhombus ABCD, AB = 2x + 3 and BC = x + 7, find CD.

Use the diagram to answer the following questions.

4.  If mÐFJH = 82, find mÐKHJ.

5.  If GH = x + 9 and JH = 5x – 2, find x.

6.  If FK = 5 and FG = 13, find KG.

7.  If mÐJFK = 6y + 7 and mÐKFG = 9y – 5, find y.

8.  If FJ = 8x – 5 and JH = 6x + 3, find GH.

9.  Quadrilateral JKLM has vertices J(-7, -2), K(0, 4), L(9, 2), and M(2, -4). Prove that quadrilateral JKLM is a rhombus. Hint: First prove that it is a parallelogram.

10. Quadrilateral TEAM has vertices T(-2, 3), E(-5, -4), A(2, -1), and M(5, 6). Prove that quadrilateral TEAM is a rhombus.

Square

1.  In square EFGH, mÐF = 2(6x + 3). Find the value of x.

2.  In square ABCD, BC = 3x - 5 and CD = 2x + 10. What is the value of BC?

3.  RSTU is a square. If SV = 5, find:

a. TR

b. RU

c. mÐSVT

d. mÐRTU

4.  In square ABCD, AB = 4x + 6 and CD = 2x + 12. What is the perimeter of the square?

To show:

Ø  Congruent sides, use ______formula.

Ø  Parallel sides, use ______formula. Parallel lines have the _____ slope.

Ø  Right Angles/Perpendicular Lines, use ______formula. Perpendicular lines have ______.

5.  Quadrilateral MATH has vertices M(-1, 1), A(4, 1), T(4, 6), H(-1, 6). Prove that it is a square.

D9 HW

1.  Quadrilateral ABCD has vertices A(–4, -5), B(1, -5), C(-2, -1), and D(-7, -1). Prove that quadrilateral ABCD is a rhombus.

2.  Quadrilateral ABCD has vertices A(3, 5), B(3, 1), C(-1, 1), and D(-1, 5). Prove that quadrilateral ABCD is a square.

1.  In isosceles trapezoid, TRAP. TR is parallel to AP. If mÐT = 73. Find all the other angles possible.

2.  In isosceles trapezoid, TRAP. TR is parallel to AP. If mÐA = 123. Find all the other angles possible.

3.  Isosceles Trapezoid ABCD with bases AB and CD. BC = 27 and mÐA = 83. Find as many other lengths and angle measures as possible.

4.  In isosceles trapezoid DUKE, diagonal DK = 10. What is the measure of diagonal UE?

5.  In isosceles trapezoid FGHJ, diagonals FH and GJ intersect at K. If FK = 8 and JG = 19. What is the measure of KH?

6.  In the accompanying diagram, isosceles trapezoid CDEF has bases of lengths 6 and 12 and an altitude of length 4. Find CD.

7.  In the accompanying diagram, isosceles trapezoid SNOW has bases of lengths 5 and 11. If SW = 7, find SY to the nearest tenth.

.

8.  The vertices of quadrilateral TRAP are T(0, 4), R(0, –8), A(3, –4), and P(3, 1). Prove that TRAP is a trapezoid.

9.  The coordinates of quadrilateral DRAW are D(–3, 6), R(3, 6), A(6, –2), and W(–6, –2). Prove that DRAW is an isosceles trapezoid.

10. Quadrilateral XMAS has coordinates X(–3, –2), M(9, 2), A(1, 6), S(–5, 4). Using coordinate geometry, prove that quadrilateral ARON is a trapezoid that contains a right angle.

D10 HW

1.  Quadrilateral ABCD has vertices A(1, 3), B(5, 3), C(8, -4), and D(-4, -4). Prove that quadrilateral ABCD is a trapezoid.

2.  Quadrilateral ABCD has vertices A(-3, -1), B(-1, 3), C(2, 3), and D(4, -1). Prove that quadrilateral ABCD is an isosceles trapezoid.

Midsegment = (Base1 + Base2)

Given the figure below, determine the missing value.

1. 2.

3. 4.

2. The coordinates of triangle MACK are M(–3, 4),

A(2, 5), C(3, 3) and K(–1, 0). Prove the MACK is an isosceles trapezoid.

1.

Practice:

1.

2. The coordinates of triangle MACK are M(–3, 4),

A(2, 5), C(3, 3) and K(–1, 0). Prove the MACK is an isosceles trapezoid.

4.

Practice:

1. A polygon is classified by its number of sides. This is the same as the number of:

(1) interior angles (2) vertices

(3) both of the above (4) neither of the above

3. A regular polygon is:

(1) equilateral but not necessarily equiangular

(2) Both equilateral and equiangular

(3) Equiangular but not necessarily equilateral

(4) not necessarily equilateral or equiangular

4. Pairs of interior and exterior angles in a polygon are:

(1) always equal (2) never equal

(3) complementary (4) supplementary

5. The sum of interior angles in a polygon is 2,700 degrees. How many sides does this polygon have?

6. How many sides does a regular polygon have if the measure of an exterior angle is 36°?

7. The sum of the measures of the interior angles of a polygon is 1440°. Name the polygon.

8. If the sum of the interior angles of a regular polygon is 720°, what is the measure of one exterior angle?

Find the value of x in the problems below.

9. 10.

x 2x 5x

3x 4x 3x + 10 2x – 5

2x 6x – 5

D C

A B

Opposite Sides are sides that do not touch. AB and CD are examples of opposite sides.

Opposite Vertices are vertices that are not connected by a side. A and C are examples of opposite vertices.

Consecutive Angles are angles whose vertices are next to each other either clockwise or counterclockwise. <ABC and <BCD are examples of consecutive angles.