91Ó°ÊÓ

If the two sides of a triangle are congruent then the angles opposite to those sides are congruent.

Consider $\mathrm{Δ}ABC$, in which$ABâ‰\dots AC$ then by isosceles triangle theorem, $∠1â‰\dots ∠2$, that is,

$m∠1=m∠2=23$.

The sum of measures of all the angles of a triangle is 180.

Consider $\mathrm{Δ}ABC$, then by an angle sum theorem,

$$\begin{gathered} m∠1+m∠2+m∠3=180 \\ 23+23+m∠3=180 \\ m∠3=180−46 \\ =134 \end{gathered}$$

As $ABâˆ\yen DE$, the interior angles are supplementary, that is,

$$\begin{gathered} m∠3+m∠4=180 \\ 134+m∠4=180 \\ m∠4=180−134 \\ =46 \end{gathered}$$

Consider $\mathrm{Δ}DCE$, in which$DEâ‰\dots DC$ then by isosceles triangle theorem, $∠5â‰\dots ∠6$, that is, $m∠5=m∠6$.

Consider $\mathrm{Δ}DCE$, then by an angle sum theorem,

$$\begin{gathered} m∠4+m∠5+m∠6=180 \\ 46+m∠5+m∠5=180 \\ 2m∠5=180−46 \\ =134 \\ m∠5=67 \end{gathered}$$

Therefore, $m∠6=m∠5=67$.

As line$AD$ is a straight line then,

$$\begin{gathered} m∠2+m∠5+m∠7=180 \\ 23+67+m∠7=180 \\ m∠7=180−90 \\ =90 \end{gathered}$$

Therefore, the measure of$∠7$ is 90.

If the two sides of a triangle are congruent then the angles opposite to those sides are congruent.

Consider $\mathrm{Δ}ABC$, in which$ABâ‰\dots AC$then by isosceles triangle theorem, $∠1â‰\dots ∠2$, that is,

width="122">m∠1=m∠2=k.

The sum of measures of all the angles of a triangle is 180.

Consider $\mathrm{Δ}ABC$, then by an angle sum theorem,

$$\begin{gathered} m∠1+m∠2+m∠3=180 \\ k+k+m∠3=180 \\ m∠3=180−2k \end{gathered}$$

As $ABâˆ\yen DE$, the interior angles are supplementary, that is,

$$\begin{gathered} m∠3+m∠4=180 \\ 180−2k+m∠4=180 \\ m∠4=180−180+2k \\ =2k \end{gathered}$$

Consider $\mathrm{Δ}DCE$, in which$DEâ‰\dots DC$ then by isosceles triangle theorem, $∠5â‰\dots ∠6$, that is, $m∠5=m∠6$.

Consider $\mathrm{Δ}DCE$, then by an angle sum theorem,

$$\begin{gathered} m∠4+m∠5+m∠6=180 \\ 2k+m∠5+m∠5=180 \\ 2m∠5=180−2k \\ m∠5=90−k \end{gathered}$$

Therefore, $m∠6=m∠5=90−k$.

As line$AD$ is a straight line then,

$$\begin{gathered} m∠2+m∠5+m∠7=180 \\ k+90−k+m∠7=180 \\ m∠7=180−90 \\ =90 \end{gathered}$$

Therefore, the measure of$∠7$ is 90.