Question

determine the value of (x) in the following diagram:

Explanation:

Identify the geometric properties

We are given a geometric figure with a central ray dividing an angle. Let the bottom vertex be \(P\), the top vertex on the central ray be \(A\), and the points on the left and right rays where perpendiculars are drawn be \(B\) and \(C\) respectively.

• The segments \(AB\) and \(AC\) are perpendicular to the outer rays \(PB\) and \(PC\), so \(\angle ABP = \angle ACP = 90^\circ\).
• The tick marks indicate that the perpendicular segments are equal in length: \(AB = AC\).
• The segment \(AP\) is shared between the two right triangles \(\triangle ABP\) and \(\triangle ACP\).

Prove triangle congruence

Using the Triangle Congruence concept, we can show that the two right triangles are congruent.

• Both \(\triangle ABP\) and \(\triangle ACP\) are right triangles because \(\angle ABP = \angle ACP = 90^\circ\).
• They share a common hypotenuse, \(AP = AP\).
• They have congruent legs, \(AB = AC\).
• By the Hypotenuse-Leg (HL) congruence theorem, \(\triangle ABP \cong \triangle ACP\).

Apply corresponding parts of congruent triangles

Using Corresponding Parts of congruent triangles:

• Since \(\triangle ABP \cong \triangle ACP\), their corresponding angles are equal.
• Therefore, the angles at the bottom vertex \(P\) are equal: \(\angle APB = \angle APC = 30^\circ\).
• Similarly, the angles at the top vertex \(A\) are equal: \(\angle BAP = \angle CAP\).

Calculate the target angle using the angle sum

Using the Triangle Angle Sum Theorem in the right-angled triangle \(\triangle ACP\):

• The sum of angles in \(\triangle ACP\) is \(180^\circ\).
• \(\angle ACP + \angle APC + \angle CAP = 180^\circ\)
• \(90^\circ + 30^\circ + \angle CAP = 180^\circ\)
• \(\angle CAP = 180^\circ - 120^\circ = 60^\circ\)

Solve for x

Since \(\angle CAP\) is represented by the expression \((7x - 3)^\circ\):

• \(7x - 3 = 60\)
• \(7x = 63\)
• \(x = 9\)

Answer:

Determine the value of \(x\) in the following diagram: <blank>\(9\)</blank>