11.03 Properties of rectangles, rhombuses, and squares
Interactive practice questions
Given the proof below, which of the following statements is shown to be true?
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Given $ABCD$ABCD is a rhombus, prove that $\angle1\cong\angle2\cong\angle3\cong\angle4$∠1≅∠2≅∠3≅∠4 and $\angle5\cong\angle6\cong\angle7\cong\angle8$∠5≅∠6≅∠7≅∠8.
A rhombus is illustrated with two diagonals intersecting at point $E$E. Rhombus $ABCD$ABCD has it vertices $A$A, $B$B, $C$C, and $D$D in a clockwise position. The diagonals bisect the angles in each vertex of the rhombus creating two adjacent angles. angle 1 and angle 2 are located at vertex $B$B. angle 3 and angle 4 are located at vertex $D$D. angle 5 and angle 6 are located at vertex $A$A. angle 8 and angle 7 are located at vertex $C$C. |
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| Statements | Reasons |
| $ABCD$ABCD is a rhombus | Given |
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$ABCD$ABCD is a parallelogram $\overline{AB}\cong\overline{BC}\cong\overline{CD}\cong\overline{DA}$AB≅BC≅CD≅DA |
Definition of a rhombus |
| $\overline{AE}\cong\overline{EC}$AE≅EC and $\overline{DE}\cong\overline{EB}$DE≅EB | If a quadrilateral is a parallelogram, then its diagonals bisect each other. |
| $\overline{AE}\cong\overline{AE}$AE≅AE, $\overline{BE}\cong\overline{BE}$BE≅BE, $\overline{CE}\cong\overline{CE}$CE≅CE, and $\overline{DE}\cong\overline{DE}$DE≅DE | Reflexive property of congruence |
| $\triangle AEB\cong\triangle CEB\cong\triangle CED\cong\triangle AED$△AEB≅△CEB≅△CED≅△AED | Side-side-side congruence theorem |
| $\angle1\cong\angle2\cong\angle3\cong\angle4$∠1≅∠2≅∠3≅∠4 and $\angle5\cong\angle6\cong\angle7\cong\angle8$∠5≅∠6≅∠7≅∠8 | Corresponding parts of congruent triangles are congruent (CPCTC). |
If a quadrilateral is a rhombus, then its diagonals bisect the angles at the vertices.
If a quadrilateral is a rhombus, then its diagonals are congruent.
If a quadrilateral has diagonals which bisect the angles at the vertices, then it is a rhombus.
If a quadrilateral is a parallelogram, then its diagonals bisect the angles at the vertices.
Given the proof below, select the correct statement and reason.
Given the proof below, which of the following statements is shown to be true?
Given the proof below, select the correct statement and reason.