your turn

use the rhombus vwxy from example 3 to find each measure.

1. $m\angle vyx$

2. $m\angle xyz$

Question

your turn

use the rhombus vwxy from example 3 to find each measure.

1. $m\angle vyx$

2. $m\angle xyz$

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Rhombus Properties", "Solving Linear Equations" ], "new_concepts": [], "current_concepts": [ "Rhombus Properties", "Solving Linear Equations", "Rhombus Theorems" ] } ## Find the value of m Using the *Rhombus Properties* and *Solving Linear Equations* knowledge points $$ \begin{aligned} VW &= WX \ 6m - 12 &= 4m + 4 \ 2m &= 16 \ m &= 8 \end{aligned} $$ ## Find the value of n Using the *Rhombus Properties* and *Solving Linear Equations* knowledge points $$ \begin{aligned} m\angle VZW &= 90^\circ \ 3n^2 - 0.75 &= 90 \ 3n^2 &= 90.75 \ n^2 &= 30.25 \ n &= 5.5 \quad (n > 0) \end{aligned} $$ ## Calculate the measure of angle VYX Using the *Rhombus Properties* knowledge point $$ \begin{aligned} m\angle WVZ &= 9n + 4 = 9(5.5) + 4 = 53.5^\circ \ m\angle VYX &= m\angle VWX = 2 \cdot m\angle WVZ = 2 \cdot 53.5^\circ = 107^\circ \end{aligned} $$ ## Calculate the measure of angle XYZ Using the *Rhombus Properties* knowledge point $$ \begin{aligned} m\angle VYX + m\angle WXY &= 180^\circ \ 107^\circ + m\angle WXY &= 180^\circ \ m\angle WXY &= 73^\circ \ m\angle XYZ &= m\angle XZW = 90^\circ \quad (\text{diagonals are perpendicular}) \ \text{Alternatively, } m\angle XYZ &= \frac{1}{2} m\angle VYX = \frac{107^\circ}{2} = 53.5^\circ \end{aligned} $$ Let's verify the angle bisector property of a rhombus: The diagonals of a rhombus bisect its angles. Therefore, diagonal $WY$ bisects $\angle VYX$. Thus, $m\angle XYZ = m\angle VYZ = \frac{1}{2} m\angle VYX = 53.5^\circ$. ## Question 1 $107^\circ$ ## Question 2 $53.5^\circ$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Rhombus Properties" ] }

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QUESTION IMAGE

Question

Explanation:

Find the value of m

Find the value of n

Calculate the measure of angle VYX

Calculate the measure of angle XYZ

Answer:

Question 1

Question 2