Given triangleABC ~ triangleDEF, m∠A = 52 degrees and m∠e = 11 degrees , what is the measure of angle C?

Answer 1

Since triangle ABC ~ triangle DEF, the corresponding angles of the triangles are equal. Therefore, the measures of angles in triangle ABC correspond to the measures of angles in triangle DEF.

Given:

• $m\mathrm{\angle }A={52}^{\circ }$
• $m\mathrm{\angle }E={11}^{\circ }$

To find $m\mathrm{\angle }C$, we first need to determine the measure of angle $B$ in triangle ABC:

The sum of angles in a triangle is always 180 degrees, so we can express the angles in triangle ABC as follows:

$$m\mathrm{\angle }A+m\mathrm{\angle }B+m\mathrm{\angle }C={180}^{\circ }$$

To find $m\mathrm{\angle }B$, we need to find the angles $D$ and $F$ in triangle DEF that correspond to angles $B$ and $C$ in triangle ABC, respectively.

Since the triangles are similar (ABC ~ DEF), we can set up the equation based on corresponding angles:

$$m\mathrm{\angle }B=m\mathrm{\angle }D={180}^{\circ }-(m\mathrm{\angle }A+m\mathrm{\angle }E)$$

Substituting the known values:

$$m\mathrm{\angle }B={180}^{\circ }-({52}^{\circ }+{11}^{\circ })$$ $$m\mathrm{\angle }B={180}^{\circ }-{63}^{\circ }={117}^{\circ }$$

Now we substitute $m\mathrm{\angle }A$ and $m\mathrm{\angle }B$ back into the triangle sum for triangle ABC to find $m\mathrm{\angle }C$:

$$m\mathrm{\angle }C={180}^{\circ }-(m\mathrm{\angle }A+m\mathrm{\angle }B)$$ $$m\mathrm{\angle }C={180}^{\circ }-({52}^{\circ }+{117}^{\circ })$$ $$m\mathrm{\angle }C={180}^{\circ }-{169}^{\circ }={11}^{\circ }$$

Thus, the measure of angle $C$ is ${11}^{\circ }$.