Question

Find the angle B and the value of z if ABC=DEF

Answer 1

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion.

The question provides that ΔABC and ΔDEF are similar. By the definition of similar triangles, we have that:

`\begin{gathered} \angle A\cong\angle D \\ \angle B\cong\angle E \\ \angle C\cong\angle F \end{gathered}`

Since congruent angles are equal, we have:

`\begin{gathered} m\angle A=m\angle D \\ m\angle B=m\angle E \\ m\angle C=m\angle F \end{gathered}`

From the provided diagram, we have the measures of the following angles:

`\begin{gathered} m\angle A=105\degree \\ m\angle E=(10z-5)\degree \\ m\angle F=30\degree \end{gathered}`

Comparing the congruent angles, we have that:

`m\angle D=105\degree`

The sum of angles in a triangle is 180°. If we add up the angles of ΔDEF, we can get the value of z:

`\begin{gathered} 105+10z-5+30=180 \\ 10z=180+5-105-30 \\ 10z=50 \\ z=5 \end{gathered}`

Given the value of z, we can get the measure of
`\begin{gathered} m\angle E=10z-5 \\ m\angle E=10(5)-5=50-5=45\degree \end{gathered}`By comparing the congruent angles, we have that:

`m\angle B=m\angle E=45\degree`

ANSWER:

Angle B = 45 degrees.

The value of z is 5.