Lesson 6 Claims and Conjectures Solidify Understanding

Ready

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.

a venn diagram with 4 right triangles in the center Figure 1

1.

, , , and are right angles.

2.

3.

4.

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?

a venn diagram with 4 right triangles in the center Figure 1

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.

6.

Quadrilateral TRAV with diagonal RA

Given:

Conjecture:

7.

Quadrilateral ABDE with Diagonals EB and AD intersecting at Point C.

Given:

Conjecture:

8.

Quadrilateral ARHC with diagonals RC and AH intersecting at Point E.

Given: and , are isosceles triangles.

Conjecture:

9.

Quadrilateral ABCD with diagonal BC. Line segment and BC and CD are marked with two tics. Line segment AC and AB are marked with one tic.

Given: and are isosceles triangles.

Conjecture:

10.

Quadrilateral ACRH with diagonals AH and RC that intersect at Point E. Line segments AE, EC, EH, and ER are marked with one tic. angle AEC and REH are marked with as right angles.

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 coordinate plane with x- and y-axis of 1-unit increments. An absolute value function with vertex (1,2). x555y555000

a.

Equation using absolute value notation

b.

Equation as a piecewise-defined function

12.

A coordinate plane with x- and y-axis of 1-unit increments. An absolute value function with vertex (2,0). x–5–5–5555y555000

a.

Equation using absolute value notation

b.

Equation as a piecewise-defined function

13.

A coordinate plane with x- and y-axis of 1-unit increments. An absolute value function with vertex (-3,0). x–5–5–5y555000

a.

Equation using absolute value notation

b.

Equation as a piecewise-defined function

14.

A coordinate plane with x- and y-axis of 1-unit increments. An absolute value function with vertex (-5,-4). x–10–10–10–5–5–5y–5–5–5000

a.

Equation using absolute value notation

b.

Equation as a piecewise-defined function