explain 2 proving diagonals of a rhombus are perpendicular
read the section below and complete the following activity (adapted from lesson 9.3).
a rhombus is a quadrilateral with four congruent sides.
the figure shows rhombus jklm.

properties of rhombuses
- if a quadrilateral is a rhombus, then it is a parallelogram.
- if a parallelogram is a rhombus, then its diagonals are perpendicular.
- if a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

example
prove that the diagonals of a rhombus are perpendicular. use the statement & reason bank below to help guide your understanding.
given: jklm is a rhombus.
prove: $jl \perp mk$

since jklm is a rhombus, $jm \cong$ ____. because jklm is also a parallelogram, $mn \cong kn$ because ____. by the reflexive property of congruence, $jn \cong jn$, so $\delta jnm \cong \delta jnk$ by the ____. so, ____ by cpctc.
by the linear pair theorem, $\angle jnm$ and $\angle jnk$ are supplementary. this means that $m\angle jnm + m\angle jnk =$ ____. since the angles are congruent, $m\angle jnm = m\angle jnk$ so by ____, $m\angle jnk + m\angle jnk = 180^\circ$ or $2m\angle jnk = 180^\circ$. therefore, $m\angle jnk =$ ____ and ____ $\perp mk$.

statement & reason bank
1. $jk$
2. diagonals of a parallelogram bisect each other
3. $jl$
4. $\angle jnm \cong \angle jnk$
5. $180^\circ$
6. substitution
7. $sss$ triangle congruence theorem
8. $90^\circ$

Question

explain 2 proving diagonals of a rhombus are perpendicular
read the section below and complete the following activity (adapted from lesson 9.3).
a rhombus is a quadrilateral with four congruent sides.
the figure shows rhombus jklm.

properties of rhombuses
- if a quadrilateral is a rhombus, then it is a parallelogram.
- if a parallelogram is a rhombus, then its diagonals are perpendicular.
- if a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

example
prove that the diagonals of a rhombus are perpendicular. use the statement & reason bank below to help guide your understanding.
given: jklm is a rhombus.
prove: $jl \perp mk$

since jklm is a rhombus, $jm \cong$ ____. because jklm is also a parallelogram, $mn \cong kn$ because ____. by the reflexive property of congruence, $jn \cong jn$, so $\delta jnm \cong \delta jnk$ by the ____. so, ____ by cpctc.
by the linear pair theorem, $\angle jnm$ and $\angle jnk$ are supplementary. this means that $m\angle jnm + m\angle jnk =$ ____. since the angles are congruent, $m\angle jnm = m\angle jnk$ so by ____, $m\angle jnk + m\angle jnk = 180^\circ$ or $2m\angle jnk = 180^\circ$. therefore, $m\angle jnk =$ ____ and ____ $\perp mk$.

statement & reason bank
1. $jk$
2. diagonals of a parallelogram bisect each other
3. $jl$
4. $\angle jnm \cong \angle jnk$
5. $180^\circ$
6. substitution
7. $sss$ triangle congruence theorem
8. $90^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Parallelogram Diagonal Properties", "Linear Pair Postulate", "Corresponding Parts of Congruent Triangles" ], "new_concepts": [ "Rhombus Diagonal Proof", "SSS Triangle Congruence Theorem" ], "current_concepts": [ "Parallelogram Diagonal Properties", "Linear Pair Postulate", "Corresponding Parts of Congruent Triangles", "Rhombus Diagonal Proof", "SSS Triangle Congruence Theorem" ] } ## Identify given information and target blanks We need to fill in the blanks of the proof showing that the diagonals of a rhombus are perpendicular. The given rhombus is $JKLM$ with diagonals intersecting at $N$. We want to prove $JL \perp MK$. The proof text with blanks is: 1. "Since $JKLM$ is a rhombus, $JM \cong$ [Blank 1]." 2. "Because $JKLM$ is also a parallelogram, $MN \cong KN$ because [Blank 2]." 3. "By the Reflexive Property of Congruence, $JN \cong JN$." (This line has a blank line before it, which seems to be a formatting separator or part of the text, but the statement is complete. Let's look closely at the image: "_____________________. By the Reflexive Property of Congruence, $JN \cong JN$." The bank has "8. $JN$" or "8. $JL$". Let's check the bank: 1 is $JK$, 2 is "Diagonals of a parallelogram bisect each other", 3 is "SSS Triangle Congruence Theorem", 4 is $\angle JNM \cong \angle JNK$, 5 is $180^\circ$, 6 is "Substitution", 7 is $90^\circ$, 8 is $JL$. Wait, let's match the blanks to the numbered items in the "Statement & Reason Bank": - "Since $JKLM$ is a rhombus, $JM \cong$ [Blank 1]." Since a rhombus has four congruent sides, $JM \cong JK$. Item 1 in the bank is $JK$. - "Because $JKLM$ is also a parallelogram, $MN \cong KN$ because [Blank 2]." The reason is "Diagonals of a parallelogram bisect each other", which is Item 2 in the bank. - "_____________________. By the Reflexive Property of Congruence, $JN \cong JN$." This line has a blank line. Let's look at the bank: Item 8 is $JL$. Wait, the line is "_____________________. By the Reflexive Property of Congruence, $JN \cong JN$." No, the line is "_____________________. By the Reflexive Property of Congruence, $JN \cong JN$." Wait, is it a blank for a statement? No, the line is just a blank line, or maybe it's asking for a statement? Let's check the bank items: 1: $JK$ 2: Diagonals of a parallelogram bisect each other 3: SSS Triangle Congruence Theorem 4: $\angle JNM \cong \angle JNK$ 5: $180^\circ$ 6: Substitution 7: $90^\circ$ 8: $JL$ Let's read the proof text carefully: "Since $JKLM$ is a rhombus, $JM \cong$ [Blank A]. Because $JKLM$ is also a parallelogram, $MN \cong KN$ because [Blank B]." "_____________________. By the Reflexive Property of Congruence, $JN \cong JN$." Wait, is the blank before "By the Reflexive Property..." actually asking for something? No, it's a line, but let's look at the next line: "so $\triangle JNM \cong \triangle JNK$ by the [Blank C]." This must be the "SSS Triangle Congruence Theorem" (Item 3). "So, [Blank D] by CPCTC." By CPCTC, the corresponding angles are congruent: $\angle JNM \cong \angle JNK$ (Item 4). "By the Linear Pair Theorem, $\angle JNM$ and $\angle JNK$ are supplementary. This means that $m\angle JNM + m\angle JNK =$ [Blank E]." Since they are supplementary, their sum is $180^\circ$ (Item 5). "Since the angles are congruent, $m\angle JNM = m\angle JNK$ so by [Blank F], $m\angle JNK + m\angle JNK = 180^\circ$ or $2m\angle JNK = 180^\circ$." This is "Substitution" (Item 6). "Therefore, $m\angle JNK =$ [Blank G] and [Blank H] $\perp MK$." $m\angle JNK = 90^\circ$ (Item 7), and $JL \perp MK$ (Item 8). ## Fill each blank using the bank Using the *Parallelogram Diagonal Properties* knowledge point - Blank 1: $JK$ (Item 1) - Blank 2: "Diagonals of a parallelogram bisect each other" (Item 2) Using the *Corresponding Parts of Congruent Triangles* knowledge point - Blank 3 (after $\triangle JNM \cong \triangle JNK$ by the): "SSS Triangle Congruence Theorem" (Item 3) - Blank 4 (So, ___ by CPCTC): $\angle JNM \cong \angle JNK$ (Item 4) Using the *Linear Pair Postulate* knowledge point - Blank 5 ($m\angle JNM + m\angle JNK =$): $180^\circ$ (Item 5) - Blank 6 (so by ___): "Substitution" (Item 6) - Blank 7 ($m\angle JNK =$): $90^\circ$ (Item 7) - Blank 8 (___ $\perp MK$): $JL$ (Item 8) Read the section below and complete the following activity (adapted from Lesson 9.3). A rhombus is a quadrilateral with four congruent sides. The figure shows rhombus $JKLM$. **Properties of Rhombuses** - If a quadrilateral is a rhombus, then it is a parallelogram. - If a parallelogram is a rhombus, then its diagonals are perpendicular. - If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. **Example:** Prove that the diagonals of a rhombus are perpendicular. Use the statement & reason bank below to help guide your understanding. Given: $JKLM$ is a rhombus. Prove: $JL \perp MK$ Since $JKLM$ is a rhombus, $JM \cong$ $JK$. Because $JKLM$ is also a parallelogram, $MN \cong KN$ because Diagonals of a parallelogram bisect each other. By the Reflexive Property of Congruence, $JN \cong JN$. so $\triangle JNM \cong \triangle JNK$ by the SSS Triangle Congruence Theorem. So, $\angle JNM \cong \angle JNK$ by CPCTC. By the Linear Pair Theorem, $\angle JNM$ and $\angle JNK$ are supplementary. This means that $m\angle JNM + m\angle JNK =$ $180^\circ$. Since the angles are congruent, $m\angle JNM = m\angle JNK$ so by Substitution, $m\angle JNK + m\angle JNK = 180^\circ$ or $2m\angle JNK = 180^\circ$. Therefore, $m\angle JNK =$ $90^\circ$ and $JL$ $\perp MK$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Properties of Rhombuses" ] }

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QUESTION IMAGE

Question

Explanation:

Identify given information and target blanks

Answer: