Question

these figures are congruent. what is m∠i?
□°

Explanation:

Step1: Recall Congruent Polygons Property

Congruent polygons have equal corresponding angles and sides. So, the sum of interior angles of a quadrilateral is \( (4 - 2)\times180^\circ= 360^\circ \). Wait, no, wait: Wait, the first figure: let's check the number of sides. Wait, the first figure: J, I, H, K, J? Wait, no, J, I, H, K, J? Wait, no, the first figure: J to I to H to K to J? Wait, no, the first figure: J, I, H, K, J? Wait, no, let's count the vertices: J, I, H, K, J? Wait, no, J, I, H, K, J: that's a quadrilateral? Wait, no, J (14 mi to I), I (47 mi to H), H (35 mi to K), K (31 mi to J), and J (125°), I (angle we need), H (56°), K (95°). Wait, no, the first figure: vertices J, I, H, K. Wait, J to I: 14 mi, I to H: 47 mi, H to K: 35 mi, K to J: 31 mi. Angles: at J: 125°, at K: 95°, at H: 56°, at I:? Wait, no, the second figure is a quadrilateral: O, N, Q, P, O? Wait, O to N: 31 mi, N to Q:?, Q to P:?, P to O:?. Angle at P: 84°. Since they are congruent, corresponding sides and angles are equal. Wait, the side J-K is 31 mi, and O-P? Wait, no, the side O-N is 31 mi, which should correspond to J-K (31 mi). So the angle at I (∠I) should correspond to the angle at P? Wait, no, wait: let's list the sides. First figure: JI = 14 mi, IH = 47 mi, HK = 35 mi, KJ = 31 mi. Angles: ∠J = 125°, ∠K = 95°, ∠H = 56°, ∠I =? Second figure: ON = 31 mi, NQ =?, QP =?, PO =?. Angle at P = 84°. Since congruent, the sides: KJ (31 mi) corresponds to ON (31 mi). So the angle at I (∠I) should correspond to the angle at P? Wait, no, wait: sum of interior angles of a quadrilateral is \( 360^\circ \). Wait, first figure: angles at J: 125°, K: 95°, H: 56°, and I:? Wait, no, wait: the first figure: is it a quadrilateral? Wait, J, I, H, K: four vertices. So sum of angles: \( \angle J + \angle I + \angle H + \angle K = 360^\circ \)? Wait, no, wait: \( 125^\circ + \angle I + 56^\circ + 95^\circ = 360^\circ \)? Wait, no, that would be \( 125 + 56 + 95 = 276 \), so \( \angle I = 360 - 276 = 84^\circ \). Wait, because the second figure has an angle at P of 84°, and since they are congruent, the corresponding angle (∠I) should be equal to ∠P (84°). Let's verify: sum of angles in quadrilateral is \( 360^\circ \). So \( 125 + 95 + 56 + x = 360 \). \( 125 + 95 = 220 \), \( 220 + 56 = 276 \), \( 360 - 276 = 84 \). So \( m\angle I = 84^\circ \).

Step1: Sum of Interior Angles of Quadrilateral

The sum of interior angles of a quadrilateral is \( (4 - 2)\times180^\circ = 360^\circ \).

Step2: Calculate Known Angles Sum

In the first (congruent) quadrilateral, known angles are \( \angle J = 125^\circ \), \( \angle K = 95^\circ \), \( \angle H = 56^\circ \). Sum of these angles: \( 125^\circ + 95^\circ + 56^\circ = 276^\circ \).

Step3: Find \( m\angle I \)

Let \( m\angle I = x \). Then \( x + 276^\circ = 360^\circ \). Solving for \( x \): \( x = 360^\circ - 276^\circ = 84^\circ \). Also, since the second quadrilateral has an angle of \( 84^\circ \) (at P) and they are congruent, corresponding angles are equal, so \( m\angle I = 84^\circ \).

Answer:

\( 84 \)