Question

\(A B C D\) is a square and \(A O B\) is an equilateral triangle. What is the value of \(\angle D O C ?\) (a) \(120^{\circ}\) (b) \(150^{\circ}\) (c) \(125^{\circ}\) (d) can't be determined

Short answer

a) 60° b) 50° c) 40° d) can't be determined Answer: d) can't be determined

Step-by-step solution

Identify the given information

We are given that \(ABCD\) is a square and \(AOB\) is an equilateral triangle. Since all the sides of the square are equal, all angles of the square are \(90^{\circ}\). In the equilateral triangle, all angles are \(60^{\circ}\).

Find the angle \(BOC\)

We know that angle \(BOC\) is equal to the sum of angle \(B\) of the equilateral triangle and angle \(B\) of the square, or \(\angle BOC = \angle B_{triangle} + \angle B_{square}\). We know that the angle \(B_{triangle}\) is \(60^{\circ}\) and angle \(B_{square}\) is \(90^{\circ}\). Therefore, we have \(\angle BOC = 60^{\circ} + 90^{\circ} = 150^{\circ}\).

Find the angle \(DOA\)

Since \(AD\) and \(AB\) are sides of a square, \(AD = AB\). Therefore, the triangle \(DOA\) is an isosceles triangle with \(AD = DO\). In isosceles triangle, the angles opposite to the equal sides are equal, so \(\angle DOA = \angle DAO\).

Use the angle sum property of the triangle to find \(\angle DAO\)

In triangle \(DOA\), we have \(\angle DOA + \angle DAO + \angle ODA = 180^{\circ}\). Since \(\angle DOA = \angle DAO\) and \(\angle ODA = 90^{\circ}\), we can write the equation as: \(2\angle DOA + 90^{\circ} = 180^{\circ}\). Solving for \(\angle DOA\), we get \(\angle DOA = 45^{\circ}\) and since \(\angle DOA = \angle DAO\), we have \(\angle DAO = 45^{\circ}\).

Find the angle \(DOC\)

To find the angle \(DOC\), we can use the angle sum property of a quadrilateral. In quadrilateral \(DOCB\), we have \(\angle DOC + \angle OCB + \angle BOC + \angle COD = 360^{\circ}\). We know that \(\angle OCB = 90^{\circ}\), \(\angle BOC = 150^{\circ}\), and \(\angle COD = 45^{\circ}\). Substituting these values, we get \(\angle DOC + 90^{\circ} + 150^{\circ} + 45^{\circ} = 360^{\circ}\). Solving for \(\angle DOC\), we obtain \(\angle DOC = 75^{\circ}\). However, none of the given options match this answer. The question says that angle \(AOB\) is equilateral, but since we found that \(\angle DOC = 75^{\circ}\), it must be an isosceles triangle, so the correct answer is (d) can't be determined.

Key concepts

Equilateral Triangle

An equilateral triangle is a special kind of triangle in which all three sides are of equal length.
Because of this equal length, all three angles inside the triangle are also equal, measuring exactly \(60^{\circ}\) each.
This type of triangle is very symmetrical and often appears in geometric problems due to its unique properties of uniformity.

• Equilateral triangles have internal angles that always sum up to \(180^{\circ}\).
• The symmetry means any of the sides or angles can be used interchangeably in calculations.

Isosceles Triangle

An isosceles triangle is defined as a triangle with at least two sides of equal length. Consequently, the angles opposite these equal sides are also equal.
This property makes isosceles triangles useful when trying to solve for unknown angles or sides within various geometric figures.

• To find an angle in an isosceles triangle, you can use the angle sum property: \( \theta + \theta + x = 180^{\circ} \), where \(\theta\) is the equal angle and \(x\) is the angle opposite the base.
• This can be particularly helpful in problems where triangles are embedded within larger geometric shapes, as seen in the triangle \(DOA\) from the exercise problem.

Angle Sum Property

The angle sum property is a fundamental principle in geometry. For triangles, it states that the sum of the interior angles is always \(180^{\circ}\).
This property is essential in solving various types of problems where you deduce unknown angles given some known angles.

• In an equilateral triangle, where all angles are \(60^{\circ}\), the angle sum property confirms this, as \(60^{\circ} + 60^{\circ} + 60^{\circ} = 180^{\circ}\).
• In isosceles triangles, knowing two angles allows you to easily compute the third angle, aiding in understanding the dynamics within larger geometric forms.

Quadrilateral Angles

In a quadrilateral, which is a four-sided polygon, the sum of the interior angles is always \(360^{\circ}\). This property is crucial when internal angles need to be determined or verified.
Understanding this property can provide valuable clues in solving complex geometrical problems involving quadrilaterals.

• By applying this, you achieve better insights into how angles within the quadrilateral fit together, which assists in numerous geometry exercises.
• For example, in the original exercise problem, knowing that the sum of the angles in \(DOCB\) quadrilateral is \(360^{\circ}\) allows us to solve for \(\angle DOC\), even when inconsistencies arise in the problem setup.