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Magazine Problem 65
CRITICAL THINKING In Exercises \(65-70\) , complete each statement with always, sometimes, or never. Explain your reasoning. A square is _____ a rhombus.
Problem 66
CRITICAL THINKING In Exercises \(65-70\) , complete each statement with always, sometimes, or never. Explain your reasoning. A rectangle is ____ a square.
Problem 67
CRITICAL THINKING In Exercises \(65-70\) , complete each statement with always, sometimes, or never. Explain your reasoning. A rectangle _____ has congruent diagonals.
Problem 68
CRITICAL THINKING In Exercises \(65-70\) , complete each statement with always, sometimes, or never. Explain your reasoning. The diagonals of a square _____ bisect its angles.
Problem 70
CRITICAL THINKING In Exercises \(65-70\) , complete each statement with always, sometimes, or never. Explain your reasoning. A rectangle _____ has perpendicular diagonals.
Problem 71
USING TOOLS You want to mark off a square region for a garden at school. You use a tape measure to mark off a quadrilateral on the ground. Each side of the quadrilateral is 2.5 meters long. Explain how you can use the tape measure to make sure that the quadrilateral is a square.
Problem 72
PROVING A THEOREM Use the plan for proof below to write a paragraph proof for one part of the Rhombus Diagonals Theorem (Theorem 7.11). Given ABCD is a parallelogram. \(\overline{\mathrm{AC}} \perp \overline{\mathrm{BD}}\) Prove ABCD is a rhombus. Plan for Proof Because ABCD is a parallelogram, its diagonals bisect each other at \(X\) Use \(\overline{A C} \perp \overline{~ B D}\) to show that \(\Delta \mathrm{BXC} \cong \triangle \mathrm{DXC}\) . Then show that \(\overline{\mathrm{BC}} \cong \overline{\mathrm{DC} .}\) Use the properties of a parallelogram to show that ABCD is a rhombus.
Problem 73
PROVING A THEOREM In Exercises 73 and 74 , write a proof for part of the Rhombus Opposite Angles Theorem (Theorem 7.12\()\) . Given PQRS is a parallelogram. \(\overline{\mathrm{PR}}\) bisects \(\angle \mathrm{SPQ}\) and \(\angle \mathrm{QRS}\) \(\mathrm{SQ}\) bisects \(\angle \mathrm{PSR}\) and \(\angle \mathrm{RQP}\) . Prove \(\quad \mathrm{PQRS}\) is a rhombus.
Problem 75
ABSTRACT REASONING Will a diagonal of a square ever divide the square into two equilateral triangles? Explain your reasoning.
Problem 76
ABSTRACT REASONING Will a diagonal of a rhombus ever divide the rhombus into two equilateral triangles? Explain your reasoning.
