Use the properties of similar triangles to determine measures (cos, sin, tan, lengths, angles) Geometry

If two triangles are similar then their corresponding angles are congruent and their side lengths are proportional. Here we can see that \overline{AC} and \overline{DE} are corresponding as they are the smallest sides of ABC and DEF, respectively. Therefore, their opposite angles, namely \widehat{B} and \widehat{E} are congruent. We have:

\widehat{A}+\widehat{B}+\widehat{C}=180^\circ

85^\circ+\widehat{B}+50^\circ=180^\circ

\widehat{B} = 45^\circ

Therefore:

If two triangles are similar then their corresponding angles are congruent and their side lengths are proportional. We have:

\dfrac{AD}{AB}=\dfrac{ED}{BC}

\dfrac{a}{a+3}=\dfrac{1}{3}

a+3=3a

2a=3

a=\dfrac{3}{2}

If two triangles are similar then their corresponding angles are congruent and their side lengths are proportional. We have:

\dfrac{ED}{BC}=\dfrac{AD}{AB}

\dfrac{ED}{7.5}=\dfrac{12}{18}

ED=\dfrac{12 \times 7.5}{18}=5

Using the Pythagoras theorem:

a=\sqrt{12^2+5^2}= 13

If two triangles are similar then their corresponding angles are congruent and their side lengths are proportional. We have:

\dfrac{ED}{BC}=\dfrac{AD}{AB}

\dfrac{2}{5}=\dfrac{3}{a}

a=\dfrac{3 \times 5}{2}=7.5

If two triangles are similar then their corresponding angles are congruent and their side lengths are proportional. We have:

\dfrac{EF}{AC}=\dfrac{a}{AB}

\dfrac{13}{26}=\dfrac{a}{AB}

a=\dfrac{{AB}}{2}

Using the Pythagoras theorem:

AB=\sqrt{26^2-10^2}=24

Therefore:

If two triangles are similar then their corresponding angles are congruent and their side lengths are proportional. We have:

\dfrac{BD}{a}=\dfrac{a}{BC}

\dfrac{1}{a}=\dfrac{a}{5}

a=\sqrt{5}

If two triangles are similar then their corresponding angles are congruent and their side lengths are proportional. We have:

\dfrac{BD}{AB}=\dfrac{AB}{BC}

\dfrac{2}{AB}=\dfrac{AB}{10}

AB=\sqrt{20}

Using the Pythagoras theorem:

a=\sqrt{{10^2}-\sqrt{20}^2}=\sqrt{100-20} = \sqrt{80}