Ken uses pattern blocks to make a design. He puts 5 pattern blocks together, as shown in the diagram.

Question

Ken uses pattern blocks to make a design. He puts 5 pattern blocks together, as shown in the diagram.
The measure of  ∠OJK
  is 120°. What is the measure of  ∠PJN
 ? Enter your answer in the box.

5 rhombuses are placed next to each other with the bottom vertex of each touching to form a pattern block. Each bottom angle of each rhombus is as follows respectively: angle PJO, angle OJN, angle NJM, angle MJI, angle IJK.

(1 point)
 degrees

GPT-4o mini · Answer

To determine the measure of $ \angle PJN $, we first need to establish the relationship between the angles formed at the vertices where the rhombuses meet. Given: - The measure of $ \angle OJK $ is $ 120^\circ $. - The rhombus has angles measuring $ 60^\circ $ (at the top) and $ 120^\circ $ (at the bottom). From the configuration, where five rhombuses are placed together with their bottom vertices touching, the angle $ \angle OJK $ includes angles from two adjacent rhombuses: 1. $ \angle IJK $ is the bottom angle of the last rhombus (we know it's $ 120^\circ $). 2. The angle $ \angle MJN $ is also $ 60^\circ $, being the top angle of the nearby rhombus to the left. To find $ \angle PJN $, we can follow this reasoning: Since $ \angle OJK $ (at point $ J $) is $ 120^\circ $, it effectively forms an angle at that vertex which can be expressed as: $$ \angle OJK = \angle MJN + \angle PJN $$ Given previous information: - We know that $ \angle MJN = 60^\circ $ (bottom angle of rhombus). - Also, we have $ \angle OJK = 120^\circ $. We can set up the equation: $$ 120^\circ = 60^\circ + \angle PJN $$ Solving this for $ \angle PJN $: $$ \angle PJN = 120^\circ - 60^\circ $$ $$ \angle PJN = 60^\circ $$ Thus, the measure of $ \angle PJN $ is: $$ \boxed{60} $$ degrees.