Question

△dog~△cat
find the measure of each of the following.
m∠d = 23 degrees
m∠o = select
x = select
y = select

Explanation:

Step1: Find angle \(m\angle O\)

Since \(\triangle DOG\sim\triangle CAT\), corresponding angles are equal. In \(\triangle CAT\), we can find the third - angle. The sum of angles in a triangle is \(180^{\circ}\). In \(\triangle CAT\), if two angles are \(35^{\circ}\) and \(81^{\circ}\), then the third angle \(\angle C=180-(35 + 81)=64^{\circ}\). So \(m\angle O = 35^{\circ}\) (corresponding angles of similar triangles).

Step2: Find side - length \(x\)

For similar triangles \(\triangle DOG\sim\triangle CAT\), the ratios of corresponding sides are equal. We have the proportion \(\frac{DG}{AT}=\frac{OG}{CT}\). Substituting the given values: \(\frac{12}{6}=\frac{x}{8}\). Cross - multiply gives \(12\times8 = 6x\), then \(x=\frac{12\times8}{6}=16\).

Step3: Find side - length \(y\)

Using the proportion of corresponding sides \(\frac{DO}{CA}=\frac{DG}{AT}\). Substituting \(DO = 15\), \(DG = 12\), \(AT = 6\), we get \(\frac{15}{y}=\frac{12}{6}\). Cross - multiply: \(12y=15\times6\), then \(y=\frac{15\times6}{12}=7.5\).

Answer:

\(m\angle O = 35^{\circ}\), \(x = 16\), \(y = 7.5\)