# Lesson 6Claims and ConjecturesSolidify Understanding

Conjectures are statements which are believed to be true based on the current data or evidence. However, they have not yet been proven or disproven. To move toward logical proof, true statements that properly build on one another, along with justifications for those statements, are needed. Based on the diagram in Figure 1, provide a justification for each statement.

### 1.

, , , and are right angles.

### 5.

Sometimes when creating a proof, it is useful to add additional segments to a diagram. Amelia is conjecturing . To prove this conjecture, Amelia thinks she will need to add to the diagram in Figure 1. What would you add to the diagram? How might this addition of an auxiliary segment be used in the proof?

## Set

Based on the information in the statements or diagrams it is possible to make claims about what is likely true. The claims made are considered conjectures because they have not yet been proven or disproven. For each problem, use the diagram and statements to generate at least one conjecture about the quadrilateral. Attempt to conjecture about the most precise classification for the quadrilateral.

Given:

Conjecture:

Given:

Conjecture:

### 8.

Given: and , are isosceles triangles.

Conjecture:

### 9.

Given: and are isosceles triangles.

Conjecture:

### 10.

Given: and , are isosceles right triangles.

Conjecture:

## Go

The graph of an absolute value function is given.

1. Write the equation using absolute value notation.

2. Then write the equation as a piecewise-defined function.

### 11.

#### a.

Equation using absolute value notation

#### b.

Equation as a piecewise-defined function

### 12.

#### a.

Equation using absolute value notation

#### b.

Equation as a piecewise-defined function

### 13.

#### a.

Equation using absolute value notation

#### b.

Equation as a piecewise-defined function

### 14.

#### a.

Equation using absolute value notation

#### b.

Equation as a piecewise-defined function