Pearson Unit 1 Topic 6 Polygons and Quadrilaterals

Pearson Unit 1 Topic 6: Polygons and Quadrilaterals 6 -4: Properties of Rhombuses, Rectangles, and Squares Pearson Texas Geometry © 2016 Holt Geometry Texas © 2007

� TEKS Focus: � (5)(A) Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. � (1)(C) Select tools, including real objects, manipulatives paper and pencil, and technology as appropriate, and techniques, including mental math, estimations, and number sense as appropriate, to solve problems. � (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. � (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.

A second type of special quadrilateral is a rectangle. A rectangle is a quadrilateral with four right angles.

A rhombus is another special quadrilateral. A rhombus is a quadrilateral with four congruent sides.

A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three.

Example: 1 Is ABCD a rhombus, rectangle or square? Explain.

Example: 2 A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM. Rect. diags. KM = JL = 86 Def. of segs. diags. bisect each other Substitute and simplify.

Example: 3 Carpentry: The rectangular gate has diagonal braces. Find HJ and HK. Rect. diags. HJ = GK = 48 Def. of segs. Rect. diags. Rect. diagonals bisect each other JL = LG Def. of segs. JG = 2 JL = 2(30. 8) = 61. 6 Substitute and simplify.

Example: 4 TVWX is a rhombus. Find TV. WV = XT 13 b – 9 = 3 b + 4 10 b = 13 b = 1. 3 Def. of rhombus Substitute given values. Subtract 3 b from both sides and add 9 to both sides. Divide both sides by 10. TV = XT Def. of rhombus TV = 3 b + 4 Substitute 3 b + 4 for XT. TV = 3(1. 3) + 4 = 7. 9 Substitute 1. 3 for b and simplify.

Example: 5 TVWX is a rhombus. Find m VTZ. m VZT = 90° 14 a + 20 = 90° a=5 m VTZ = m ZTX m VTZ = (5 a – 5)° m VTZ = [5(5) – 5)]° = 20° Rhombus diag. Substitute 14 a + 20 for m VTZ. Subtract 20 from both sides and divide both sides by 14. Rhombus each diag. bisects opp. s Substitute 5 a – 5 for m VTZ. Substitute 5 for a and simplify.

Example: 6 CDFG is a rhombus. Find CD. CG = GF Def. of rhombus 5 a = 3 a + 17 Substitute a = 8. 5 Simplify GF = 3 a + 17 = 42. 5 Substitute CD = GF Def. of rhombus CD = 42. 5 Substitute

Example: 7 CDFG is a rhombus. Find m GCH if m GCD = (b + 3)° and m CDF = (6 b – 40)° m GCD + m CDF = 180° b + 3 + 6 b – 40 = 180° 7 b = 217° b = 31° Def. of rhombus Substitute. Simplify. Divide both sides by 7.

Example: 7 continued m GCH + m HCD = m GCD 2 m GCH = m GCD Rhombus each diag. bisects opp. s 2 m GCH = (b + 3) Substitute. 2 m GCH = (31 + 3) m GCH = 17° Substitute. Simplify and divide both sides by 2.

Example: 8 A carpenter’s square can be used to test that an angle is a right angle. How could the contractor use a carpenter’s square to check that the frame is a rectangle? 12 feet 8 ft 12 feet Both pairs of opp. sides of WXYZ are , so WXYZ is a parallelogram. The contractor can use the carpenter’s square to see if one of WXYZ is a right . If one angle is a right , then by Theorem 6 -5 -1 the frame is a rectangle.