16. abc is a right-angled triangle. azyx is a trapezium where ax // zy. azb is a straight line. find $\angle abc$.

17. abcd is trapezium where ad // bc. cdef is a rhombus. ade and bcf are straight lines. find $\angle def$.

18. abcd is a rhombus. bcd is an isosceles triangle with bc = cd. edc is a straight line. find $\angle adb$.

Question

16. abc is a right-angled triangle. azyx is a trapezium where ax // zy. azb is a straight line. find $\angle abc$.

17. abcd is trapezium where ad // bc. cdef is a rhombus. ade and bcf are straight lines. find $\angle def$.

18. abcd is a rhombus. bcd is an isosceles triangle with bc = cd. edc is a straight line. find $\angle adb$.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Angle Properties of Polygons", "Isosceles Triangle Properties", "Rhombus Angle Properties" ], "new_concepts": [ "Trapezium Angle Properties", "Parallel Line Angle Properties" ], "current_concepts": [ "Angle Properties of Polygons", "Isosceles Triangle Properties", "Rhombus Angle Properties", "Trapezium Angle Properties", "Parallel Line Angle Properties" ] } ## Solve Question 16 Using the *Angle Properties of Polygons* knowledge point $$ \begin{aligned} &\text{Since } AZYX \text{ is a trapezium with } AX \parallel ZY:\ &\angle XAZ + \angle YZA = 180^\circ \implies 90^\circ + \angle YZA = 180^\circ \implies \angle YZA = 90^\circ\ &\text{Since } AZB \text{ is a straight line, } \angle YZB = 180^\circ - \angle YZA = 90^\circ\ &\text{In } \triangle YZB, \text{ the sum of angles is } 180^\circ:\ &\angle ZYB = 180^\circ - \angle YZB - \angle YBZ = 180^\circ - 90^\circ - \angle ABC = 90^\circ - \angle ABC\ &\text{Given } \angle CYZ = 75^\circ \text{ and } C \text{ lies on } AX \text{ (so } \angle CYZ \text{ is } \angle X Y Z \text{ or related, but from diagram } \angle CYZ = 75^\circ):\ &\text{Since } AX \parallel ZY, \text{ alternate interior angles give } \angle ACY = \angle CYZ = 75^\circ\ &\text{In right-angled } \triangle ABC \text{ at } A:\ &\angle ABC = 90^\circ - \angle ACB = 90^\circ - 75^\circ = 15^\circ \end{aligned} $$ ## Solve Question 17 Using the *Rhombus Angle Properties* and *Angle Properties of Polygons* knowledge points $$ \begin{aligned} &\text{Since } ABCD \text{ is a trapezium with } AD \parallel BC:\ &\angle ADC = 180^\circ - \angle BCD = 180^\circ - 73^\circ = 107^\circ\ &\text{Since } ADE \text{ is a straight line, } \angle CDE = 180^\circ - \angle ADC = 73^\circ\ &\text{Since } CDEF \text{ is a rhombus, opposite angles are equal:}\ &\angle DEF = \angle CDE = 73^\circ \end{aligned} $$ ## Solve Question 18 Using the *Rhombus Angle Properties* and *Isosceles Triangle Properties* knowledge points $$ \begin{aligned} &\text{Since } EDC \text{ is a straight line, } \angle ADC = 180^\circ - \angle ADE = 180^\circ - 80^\circ = 100^\circ\ &\text{Since } ABCD \text{ is a rhombus, adjacent angles sum to } 180^\circ:\ &\angle BCD = 180^\circ - \angle ADC = 180^\circ - 100^\circ = 80^\circ\ &\text{In rhombus } ABCD, \text{ the diagonal } BD \text{ bisects } \angle ADC:\ &\angle ADB = \frac{1}{2} \angle ADC = \frac{1}{2} (100^\circ) = 50^\circ \end{aligned} $$ ## Question 16 $15^\circ$ ## Question 17 $73^\circ$ ## Question 18 $50^\circ$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Angle Properties of Polygons" ] }

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