Question

7
triangle abc with angle at a: 37°, angle at c: (13x - 5)°
triangle def with angle at f: 73°, angle at d: (6x + 1)°, angle at e: (9x + 16)°

Explanation:

Step1: Find angle B in triangle ABC

In triangle \(ABC\), we know angle \(A = 37^\circ\) and angle \(C=(13x - 5)^\circ\). The sum of angles in a triangle is \(180^\circ\), so angle \(B=180^\circ - 37^\circ-(13x - 5)^\circ=148^\circ - 13x^\circ\).

Step2: Find angle D in triangle DEF

In triangle \(DEF\), angle \(F = 73^\circ\), angle \(E=(9x + 16)^\circ\), angle \(D=(6x + 1)^\circ\). The sum of angles in a triangle is \(180^\circ\), so \(73^\circ+(9x + 16)^\circ+(6x + 1)^\circ=180^\circ\).
Simplify the left - hand side: \(73 + 9x+16 + 6x + 1=180\)
Combine like terms: \(90 + 15x=180\)
Subtract 90 from both sides: \(15x=180 - 90=90\)
Divide both sides by 15: \(x = 6\)

Step3: Check the angles

First, find angle \(C\): Substitute \(x = 6\) into \((13x - 5)^\circ\), we get \(13\times6-5=78 - 5 = 73^\circ\)
Then, find angle \(B\): \(148-13\times6=148 - 78 = 70^\circ\)
In triangle \(DEF\), angle \(D=(6\times6 + 1)^\circ=37^\circ\), angle \(E=(9\times6+16)^\circ=54 + 16 = 70^\circ\)
We can see that triangle \(ABC\) and triangle \(DEF\) have corresponding angles equal (angle \(A = 37^\circ=\) angle \(D\), angle \(B = 70^\circ=\) angle \(E\), angle \(C = 73^\circ=\) angle \(F\)), so the triangles are similar. And we found the value of \(x\) by using the angle - sum property of triangles.

Answer:

The value of \(x\) is \(\boldsymbol{6}\)