Angle Relationships

10.1

Outcomes

E7 – make and apply generalizations about angle relationships.

Always true?Sometimes true?Never true?

1. The diagonals of a parallelogram are congruent.

2. Parallelograms are rectangles.3. Vertically opposite angles are across

from each other.4. The angles in a triangle can measure

100o.5. If two rectangles have the same area,

the rectangles are congruent.6. Angles in a square are 90o.

Always true?Sometimes true?Never true?

1. The diagonals of a parallelogram are always congruent.

2. All parallelograms are rectangles.3. Vertically opposite angles are across

from each other.4. None of the angles in a triangle can

measure 100o.5. If two rectangles have the same area,

the rectangles are congruent.6. All angles in a square are 90o.

Stand up and show me using your arms…. The following angles

Acute Obtuse 90o

180o

0o

Reflex Now take two Geostrips and repeat

Supplementary or Complementary?

Supplementary angles – two angles whose sum is 180o.

Complementary angles – two angles whose sum is 90o.

Join two pattern blocks to show

A pair of supplementary angles. Sketch the pattern blocks to illustrate your

answers for this and each of the following. A pair of congruent pattern blocks. A pair of non-congruent pattern blocks. A different pair of non-congruent

pattern blocks.

Use a Power Polygon square to verify that the angles are complementary. Draw diagrams to illustrate your

answers.

Find a combination of two Power Polygons that show complementary angles.

Find two other possible solutions, using two different Power Polygons.

Vertically Opposite Angles

Vertically opposite angles are non-adjacent angles formed when two lines cross.

A

BC

ED

How are vertically opposite angles related? Join a pair of Geostrips so that they form an X,

as shown Take a pattern block of your choice. Adjust the

Geostrip model so that one of the angles is congruent to one of the angles of the pattern block.

Without changing the model, check the size of the vertically opposite angles. How are the angles related?

Investigate the other pair of opposite angles using pattern blocks.

Repeat using different pattern blocks or Power Polygons.

Method 1: Sum of the interior angles of a triangle!

Cut off the corner of your page and label each of the corners as A, B and C. Tear off the corners and put them together.

Sum of interior angles in Polygons

A + B + C = 180o

Use a straight edge to draw any quadrilateral.

Draw one diagonal in it.Explain how you can use this diagram

to find the sum of the measures of the interior angles of the quadrilateral.

V V

V

Method 2: Select three congruent triangles (other than

equilateral triangles) from the Power Polygons set. Trace one of the triangles onto paper and label them a, b, c

Place the three triangles in a way that shows that the sum of their interior angles is 180o. Record your work by tracing around the three triangles

Explain how your model demonstrates this property.

Repeat using a different triangle Does this convince you that the sum of the interior

angles of a triangle is 180o for all triangles?

Use a straight edge to draw any pentagon (does not have to be regular).

From one vertex only, draw all possible diagonals.

How many diagonals meet at this vertex? Explain how you can use this model to

find the sum of the interior angles of a pentagon. Find this sum.

Does this sum change if you change the size or shape of the pentagon?

Complete this table

Polygon Diagram Number of sides

Number of

Diagonals from one

vertex

Number of

TrianglesFormed

Sum of Interior Angles

Triangle 3 0 1 180o

Quadrilateral

4 1 2

Pentagon 5

Polygon Diagram Number of sides

Number of

Diagonals from one

vertex

Number of

TrianglesFormed

Sum of Interior Angles

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Hendecagon

n-gon

Transversal

A line that crosses two other lines or line segments

Angles Formed by Parallel Lines and a Transversal

Build the following using Geostrips. Sketch your construction into your notebook and label it as shown.

A

D

G

C

H

E

B

F

transversal

Use pattern blocks to find congruent angles in your construction. Record the congruent angles.

Use pattern blocks to find the supplementary angles.

Name pairs of angles that always appear to be congruent.

Name pairs of angles that always appear to be supplementary

Use your last sketch to answer the following.1. Which angle pair are vertically opposite? What

would CBH measure if ABG measured 40o? Why

2. Place your construction on top of your drawing. Slide the construction down so the CBG coincides with FEG of your drawing.

3. What can you say about these two angles? 4. These angles are named corresponding angles.5. Why might they be called this? 6. Work with a partner to find other pairs of

corresponding angles.

VVV

V

Corresponding Angles

Angles either both above or both below two lines on the same side of the transversal

If the lines are parallel, corresponding angles are congruent

x

y

What is the relationship between angles ABH and DEG?

Why might these angles be called interior angles? Work with a partner and using both of your constructions find other pairs of interior angles.

Interior angles

Angles between two lines on the same side of a transversal.

If the lines are parallel, interior angles are supplementary.

b

a

>

>

a + b = 180o

What is the relationship between angles ABH and FEG?

Why might these angles be called alternate interior angles?

Work with a partner and using both of your constructions find other pairs of alternate interior angles.

Alternate interior angles

Angles between two lines on either side of a transversal

If the lines are parallel, alternate interior angles are congruent.

a

b

>

>

What happens to angle relationships when a transversal crosses two lines that are not parallel?

Remove the pair of opposite sides that are not crossed by the transversal. Move one of the strips so that the lines are no longer parallel.

Examine what happens to the following angle relationships when a transversal crosses two non-parallel lines: Alternate interior angles Corresponding angles Interior angles Vertically opposite angles

Example 1: Parallel Lines and a Transversal

Find the measures of EHB, FHG, and EHGV

V V

>

>C

A G

D

E

HB

F

131o

Line segment AB is a transversal.

Example 2: Apply Angle Relationships to Find Unknown Angles

A B

D

C

E

29ox

64oF

xx

C

BA

ED

C

80o

D

F

E

BA G

z z

y

Communicate Key Ideas

Page 431 #1 and 2

Find the measure of ABC, CBD, and ECD. Explain your reasoning.

ED

C

B

A

80

35

Answer

ABC = 145o CBD = 35o ECD = 65o E

D

C

B

A

80

35

Question #2

a) Describe a method for determining the sum of the interior angles of a polygon, without having to measure them.

b) Does this method work for polygons that are not regular as well as regular polygons? Explain why or why not.

c) Suppose you know the number of sides of a polygon. Can you determine the measures of its interior angles, assuming that the polygon is regular and not regular? Explain your answer.

Answers

a) S = 180o (n-2) , where S represents the sum of the interior angles and n represents the number of sides of the polygon.

b) Yes, the sum of interior angles is the same for regular and not regular polygons

c) Yes, all the interior angles will have the same measure, so simply divide the sum of the interior angles by the numbers of sides. No, the interior angles will have different measures.

Question #3

>

>

a bc d

hgfe

a) List the corresponding angles, alternate interior angles, and interior

angles on the same side of the transversal.b) Why do you think these names make sense?

Answers

a) Corresponding angles: a and e, c and f, d and h alternate interior angles: c and f, e and d interior angles: c and e, d and f

Check Your Understanding

Page 432 – 433 #1, 2, 4, and 5

Page 432 #1a) (labelled D in textbook)

d

ee

62.0

A DG

C

E

B

F

Question 1b)

i

hh

g

f

110

R

M

Q

L N

SP

Answer

Question 1c)

z

y

x

w

c)B

F

H

D

G

I

E

C

A

J

L

K

Answer

Q.#2 Classify as always true, sometimes true, never true

a) the sum of the interior angles in a triangle is 180o

b) alternate interior angles on a transversal are congruent

c) a triangle has two right angles d) interior angles on the same side formed by an

transversals and two parallel lines are equal in measure e) a regular polygon has at least one pair of parallel

sides. f ) a quadrilateral has four acute angles. g) three of the angles of a quadrilateral can be acute h) vertically opposite angles are complementary

If you pick ‘always true’ or ‘never true’ explain how you know. If you pick ‘sometimes true’ describe the conditions necessary to make it true.

Answers

Question #4 I am a quadrilateral. I have two pairs

of congruent angles and two pairs of supplementary angles. Who am I, and how do you know? Is there more than one possible answer?

Explain, using words and diagrams. Two of my sides are equal in length, the

other two are different. Which of these quadrilaterals am I? How do you know?

Answer

#5

I am a right triangle. One of my two complementary angles is twice the measure of the other. What are the measure of my three

angles? Explain how you know. Draw me, using only pattern blocks to

measure the angles. Explain your method.