91Ó°ÊÓ

Is it possible that any triangle can be partitioned into four congruent triangles that can be rearranged to form a parallelogram? Explain your reasoning.

In Exercises \(43-48,\) the diagonals of rectangle QRST intersect at \(\mathrm{P}\) . Given that \(\mathrm{m} \angle \mathrm{PTS}\) \(34^{\circ}\) and \(\mathrm{QS}\) 10, Find the indicated measure. $$ \mathrm{m} \angle \mathrm{SRT} $$

In Exercises \(43-48,\) the diagonals of rectangle QRST intersect at \(\mathrm{P}\) . Given that \(\mathrm{m} \angle \mathrm{PTS}\) \(34^{\circ}\) and \(\mathrm{QS}\) 10, Find the indicated measure. $$ \mathrm{QP} $$

In trapezoid PQRS, \(\overline{\mathrm{PQ}} \mathrm{RS}\) and \(\overline{\mathrm{MN}}\) is the midsegment of PQRS. If \(\mathrm{RS}=5 . \mathrm{PQ}\) , what is the ratio of MN to RS? a) \(3 : 5\) b) \(5 : 3\) c) \(1 : 2\) d) \(3 : 1\)

In Exercises \(43-48,\) the diagonals of rectangle QRST intersect at \(\mathrm{P}\) . Given that \(\mathrm{m} \angle \mathrm{PTS}\) \(34^{\circ}\) and \(\mathrm{QS}\) 10, Find the indicated measure. $$ \mathrm{RT} $$

Prove the Congruent Parts of Parallel Lines Corollary : If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. graph cannot copy Given: GH, JK, LM, GJ \(\cong \overline{\mathrm{JL}}\) Prove \(\quad \overline{\mathrm{HK}} \cong \overline{\mathrm{KM}}\) (Hint: Draw \(\overline{\mathrm{KP}}\) and \(\overline{\mathrm{MQ}}\) such that quadrilateral GPKJ and quadrilateral JQML are parallelograms.)

ABSTRACT REASONING The midpoints of the sides of a quadrilateral have been joined to form what looks like a parallelogram. Show that a quadrilateral formed by connecting the midpoints of the sides of any quadrilateral is always a parallelogram. (Hint: Draw a diagram. Include a diagonal of the larger quadrilateral. Show how two sides of the smaller quadrilateral relate to the diagonal.)

In Exercises \(55-60\) , decide whether \(JKLM\) is a rectangle, a rhombus, or a square. Give all names that apply. Explain your reasoning. $$ \mathrm{J}(-4,2), \mathrm{K}(0,3), \mathrm{L}(1,-1), \mathrm{M}(-3,-2) $$

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.

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.