Lesson 1 How Do You Know That? Develop Understanding

Ready

It is said that a picture is worth a thousand words. Mathematical symbols can be just as important when communicating about geometric diagrams. Decide which of the symbolic statements are correct based on what is given in the diagram. Justify your answers.

1.

a.

$m ∠ A B C = m ∠ F D E$

b.

$m ∠ A B C ≅ m ∠ F D E$

c.

$△ A C B ≅ △ F E D$

d.

$△ A C B = △ F E D$

2.

a.

$∠ C A T = ∠ R U G$

b.

$∠ C A T ≅ ∠ R U G$

c.

$△ C A T ≅ △ R U G$

d.

$△ C A T = △ R U G$

Create a diagram based on the given symbolic statement. Be sure enough information is provided in the diagram to validate each statement.

3.

$△ F U N ≅ △ C A R$

4.

$m ∠ H A T = 50 °$

5.

$\overset{\leftrightarrow}{E T} ⊥ \overset{\leftrightarrow}{L A}$

6.

$m ∠ R A T + m ∠ T A C = 180 °$

Set

There are four “ways of knowing” that were discussed as part of today’s lesson. They are “knowing” based on: Authority, Examples, Diagrams, and Logical Reasoning.

Consider each of the scenarios below and identify which “way of knowing” best describes what is happening.

7.

Every time I have drawn a ray with its endpoint on a line and measure the two angles formed with a protractor, they add up to $180 °$ . So, adjacent angles that form a straight line are supplementary.

A.

Knowing based on authority

B.

Knowing based on examples

C.

Knowing based on diagrams

D.

Knowing based on logical reasoning

8.

You know triangles that have a right angle are classified as right triangles. And you know the sides of a right triangle have a relationship described by the Pythagorean theorem. So you have concluded that when you have a triangle with a right angle you can use the Pythagorean theorem to find missing sides.

A.

Knowing based on authority

B.

Knowing based on examples

C.

Knowing based on diagrams

D.

Knowing based on logical reasoning

9.

I know supplementary means the two angles add up to $180 °$ because my teacher from a few years ago just told me.

A.

Knowing based on authority

B.

Knowing based on examples

C.

Knowing based on diagrams

D.

Knowing based on logical reasoning

10.

We tore off the corners of a triangle and placed them together to have a straight line. Since we had everyone in our class do it we know the angles of a triangle will add up to $180 °$ .

A.

Knowing based on authority

B.

Knowing based on examples

C.

Knowing based on diagrams

D.

Knowing based on logical reasoning

11.

I know the Pythagorean Theorem only works for right triangles because I read it in a book. It said it is a relationship to only be used with right triangles.

A.

Knowing based on authority

B.

Knowing based on examples

C.

Knowing based on diagrams

D.

Knowing based on logical reasoning

12.

I know you can always turn a quadrilateral into two triangles because of the way they look. Just draw a picture of any quadrilateral and then connect opposite corners and you will have two triangles. It works because of the drawing, try it.

A.

Knowing based on authority

B.

Knowing based on examples

C.

Knowing based on diagrams

D.

Knowing based on logical reasoning

Go

If two angles share a vertex and together they make a straight angle, then the two angles are called a linear pair. (Below are 3 examples of linear pairs.)

Examples of linear pairs in real life:

A light house with A straight line divided into two angles by a ray superimposed.

a leaning house with A straight line divided into two angles by a ray superimposed.

13.

Draw at least $2$ diagrams of a real life linear pair.

For $2$ angles to be a linear pair, they must share a vertex and a side, and the sum of their measures must equal $180 °$ .

Find the measure of the missing angle.

14.

15.

16.

17.

18.

Linear pairs could be defined as being supplementary angles because they always add up to $180 °$ . Are all supplementary angles linear pairs? Explain.

Find the supplement of the given angle. Then draw the two angles as linear pairs. Label each angle with its measure.

19.

$m ∠ A B C = 72 °$

$B$ will be the vertex.

20.

$m ∠ G H K = 113 °$

$H$ will be the vertex.