1. **Problem statement:** Given isosceles trapezoid KWLN with sides and angles: - $KN=19.5$ cm - $KL=15$ cm - $m\angle KNL=88^\circ$ - $KW=20$ cm - Median $AB=12$ cm Find: 1) $NL$, 2) $m\angle WLN$, 3) $m\angle KWL$, 4) $m\angle WJN$, 5) $WN$, 6) $WL$, 7) $WB$, 8) $m\angle KAB$, 9) $m\angle ABC$, 10) $BL$. 2. **Important properties and formulas:** - In an isosceles trapezoid, the legs are equal: $KW=NL$ and $KN=WL$. - The median $AB$ is parallel to bases and equals the average of the bases: $AB=\frac{KN+WL}{2}$. - Angles adjacent to each leg are equal in isosceles trapezoid. 3. **Step 1: Find $NL$** Since trapezoid is isosceles, legs are equal: $KW=NL=20$ cm. 4. **Step 2: Find $WL$** Given $KN=19.5$ cm and $KL=15$ cm. Since $KN$ and $WL$ are legs, $WL=KN=19.5$ cm. 5. **Step 3: Verify median $AB$ length** Median formula: $$AB=\frac{KN+WL}{2} = \frac{19.5 + 19.5}{2} = \frac{39}{2} = 19.5$$ But given $AB=12$ cm, so $AB$ is not median of legs but of bases. 6. **Step 4: Find bases $KN$ and $WL$** Since $AB$ is median of bases $KN$ and $WL$, and $AB=12$ cm, $$AB=\frac{KN+WL}{2} \Rightarrow 12=\frac{KN+WL}{2} \Rightarrow KN+WL=24$$ Given $KN=19.5$, so $$WL=24 - 19.5 = 4.5 \text{ cm}$$ 7. **Step 5: Find $NL$** Since trapezoid is isosceles, legs $KW=NL=20$ cm. 8. **Step 6: Find $WN$** $WN$ is the other base, so $WN=WL=4.5$ cm. 9. **Step 7: Find $WL$** $WL=4.5$ cm (from step 6). 10. **Step 8: Find angles $m\angle WLN$ and $m\angle KWL$** Since $m\angle KNL=88^\circ$ and trapezoid is isosceles, angles at $W$ and $L$ are equal. Using triangle $KNL$, sum of angles is $180^\circ$: $$m\angle KNL + m\angle KLN + m\angle LKN = 180^\circ$$ Given $m\angle KNL=88^\circ$, and $m\angle KLN = m\angle LKN$ (isosceles triangle), Let $x = m\angle KLN = m\angle LKN$, then $$88 + 2x = 180 \Rightarrow 2x = 92 \Rightarrow x = 46^\circ$$ So $m\angle WLN = 46^\circ$ and $m\angle KWL = 46^\circ$. 11. **Step 9: Find $m\angle WJN$** Point $J$ is intersection of diagonals $KW$ and $NL$. In isosceles trapezoid, diagonals are equal and intersect symmetrically. Thus, $m\angle WJN = m\angle KAB = 46^\circ$ (same as base angles). 12. **Step 10: Find $WB$ and $BL$** Since $AB$ is median, $A$ and $B$ are midpoints of legs. $WB$ is half of $WL=4.5$ cm, so $$WB = \frac{WL}{2} = \frac{4.5}{2} = 2.25 \text{ cm}$$ Similarly, $BL$ is half of $KL=15$ cm, $$BL = \frac{KL}{2} = \frac{15}{2} = 7.5 \text{ cm}$$ **Final answers:** 1) $NL=20$ cm 2) $m\angle WLN=46^\circ$ 3) $m\angle KWL=46^\circ$ 4) $m\angle WJN=46^\circ$ 5) $WN=4.5$ cm 6) $WL=4.5$ cm 7) $WB=2.25$ cm 8) $m\angle KAB=46^\circ$ 9) $m\angle ABC=46^\circ$ 10) $BL=7.5$ cm