Question

consider the following figure.
(note that the figure is not drawn to scale.)
order the side lengths ab, bd, ac, ad, and dc from least to greatest.
□ < □ < □ < □ < □

Explanation:

Step1: Find angles in $\triangle ABD$

In $\triangle ABD$, sum of angles is $180^\circ$. So $\angle ADB = 180^\circ - 40^\circ - 77^\circ = 63^\circ$.

Step2: Find $\angle ADC$

$\angle ADC$ is supplementary to $\angle ADB$, so $\angle ADC = 180^\circ - 63^\circ = 117^\circ$.

Step3: Find $\angle DAC$ in $\triangle ADC$

In $\triangle ADC$, sum of angles is $180^\circ$. So $\angle DAC = 180^\circ - 117^\circ - 10^\circ = 53^\circ$.

Step4: Analyze $\triangle ABD$ side lengths

In $\triangle ABD$, angles are $40^\circ$ (at A), $77^\circ$ (at B), $63^\circ$ (at D). By triangle side - angle relationship (larger angle opposite longer side), sides: $AB$ (opposite $63^\circ$), $BD$ (opposite $40^\circ$), $AD$ (opposite $77^\circ$). So $BD < AB < AD$.

Step5: Analyze $\triangle ADC$ side lengths

In $\triangle ADC$, angles are $10^\circ$ (at C), $117^\circ$ (at D), $53^\circ$ (at A). Sides: $DC$ (opposite $53^\circ$), $AC$ (opposite $117^\circ$), $AD$ (opposite $10^\circ$). So $AD < DC < AC$.

Step6: Combine the order

From $BD < AB < AD$ and $AD < DC < AC$, we get $BD < AB < AD < DC < AC$.

Answer:

$BD < AB < AD < DC < AC$