Pearson Unit 1 Topic 6 Polygons and Quadrilaterals

�TEKS Focus: �(6)(E) Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using

To prove that a given quadrilateral is a square, it is sufficient to show

Example: 1 Determine if the conclusion is valid. If not, tell what additional information

Example: 2 Determine if the conclusion is valid. If not, tell what additional information

Example: 2 continued Step 2 Determine if EFGH is a rectangle. Given. EFGH is

Example: 3 Determine if the conclusion is valid. If not, tell what additional information

Example: 4 For what values of x and y is quadrilateral QRST a rhombus?

Example: 5 Can you conclude that each parallelogram is a rectangle, rhombus or a

Example: 7 For what value of x is quadrilateral ABCD a rectangle? A (5

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�TEKS Focus: �(6)(E) Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems. �(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. �(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Remember:

To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. Remember! You can also prove that a given quadrilateral is a rectangle, rhombus, or square by using the definitions of the special quadrilaterals.

Example: 1 Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: Conclusion: EFGH is a rhombus. The conclusion is not valid. You must first know that the quadrilateral is a parallelogram.

Example: 2 Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: Conclusion: EFGH is a square. Step 1 Determine if EFGH is a parallelogram. Given EFGH is a parallelogram. Quad. with diags. bisecting each other

Example: 2 continued Step 2 Determine if EFGH is a rectangle. Given. EFGH is a rectangle. with diags. rect. Step 3 Determine if EFGH is a rhombus. with one pair of cons. sides rhombus Step 4 Determine is EFGH is a square. Since EFGH is a rectangle and a rhombus, it has four right angles and four congruent sides. So EFGH is a square by definition. The conclusion is valid.

Example: 3 Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: ABC is a right angle. Conclusion: ABCD is a rectangle. The conclusion is not valid. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. To apply this, you need to know that ABCD is a parallelogram.

Example: 4 For what values of x and y is quadrilateral QRST a rhombus? 2 y = y+ 25 y = 25 4 x – 32 = 2 x + 4 2 x – 32 = 4 2 x = 36 x = 18

Example: 5 Can you conclude that each parallelogram is a rectangle, rhombus or a square? Explain. Rectangle, because diagonals are congruent Rhombus, because diagonals are perpendicular

Example: 6 m 2 = 28 m 3 = 28 m 4 = 28

Example: 7 For what value of x is quadrilateral ABCD a rectangle? A (5 x + 2) (3 x) D 5 x + 2 + 3 x = 90 8 x + 2 = 90 8 x = 88 x = 11 B C