Question

these figures are congruent. what is m∠p?

Explanation:

Step1: Recall congruent triangles property

Congruent triangles have corresponding angles equal. So we need to find the corresponding angle to ∠P in triangle LKJ.

Step2: Identify corresponding angles

First, find the angles in triangle LKJ. The angles are 43°, 76°, and 61° (since sum of angles in a triangle is 180°, 43 + 76 + 61 = 180). Now, match the sides and angles. Triangle LKJ has side KJ = 50 m, angle at K is 76°, angle at J is 61°, angle at L is 43°. Triangle OPN (wait, labels: P, O, N) has side OP = 50 m? Wait, no, triangle LKJ: KJ = 50 m, angle at K: 76°, angle at J: 61°, angle at L: 43°. Triangle PON (P, O, N): side PN = 71 m, OP = 50 m, ON = 64 m. Wait, in triangle LKJ, side KJ = 50 m (between K (76°) and J (61°)), angle at L is 43°. In triangle PON, angle at N is 43°, angle at O is 76°, side OP = 50 m (between O (76°) and P? Wait, no, let's list the angles. In triangle LKJ: angles are ∠K = 76°, ∠J = 61°, ∠L = 43°. In triangle PON: ∠N = 43°, ∠O = 76°, so the remaining angle ∠P should be equal to ∠J, because sum of angles in a triangle is 180. So 180 - 76 - 43 = 61? Wait, no: 180 - 76 (∠O) - 43 (∠N) = 61. So ∠P is 61°, because in congruent triangles, corresponding angles are equal. Wait, let's check the sides. KJ = 50 m (in LKJ) corresponds to OP = 50 m (in PON)? Wait, KJ is between K (76°) and J (61°), OP is between O (76°) and P? Wait, no, maybe the correspondence is LKJ ≅ PNO? Wait, labels: LKJ and PON. Let's see: ∠L = 43° (matches ∠N = 43°), ∠K = 76° (matches ∠O = 76°), so ∠J (61°) matches ∠P. So yes, ∠P = ∠J = 61°.

Step1: Sum of angles in a triangle

In any triangle, sum of interior angles is 180°. For triangle LKJ, angles are 43°, 76°, and 61° (43 + 76 + 61 = 180).

Step2: Corresponding angles in congruent triangles

Since the triangles are congruent, corresponding angles are equal. In triangle LKJ, ∠J is 61°. In triangle PON, ∠P corresponds to ∠J (because ∠L corresponds to ∠N (43°), ∠K corresponds to ∠O (76°), so ∠J corresponds to ∠P). Thus, m∠P = 61°.

Answer:

61