Question

these figures are congruent. what is m∠m? \boxed{}°

Explanation:

Step1: Recall Congruent Figures Property

Congruent figures have corresponding angles equal. So, we need to find the corresponding angle to ∠M in the second figure.

Step2: Identify Corresponding Angle

Looking at the side lengths: MN = 17 mi, VM = 17 mi? Wait, no, MN is 17 mi, VM (wait, in the second figure, V to W? Wait, no, let's check the sides. ON is 15 mi? Wait, O to N is 15 mi, U to V? No, U to T is 48 mi, O to L is 48 mi. L to M is 49 mi, T to W is 49 mi. N to M is 17 mi, V to W? No, V to U? Wait, N to M is 17 mi, V to W? No, V to U? Wait, the angle at W is 103°, let's check the angles. Wait, in the first figure, angles are at O (109°), N (116°), L (32°), M (unknown), and the other angle? Wait, no, it's a pentagon? Wait, no, wait the figures: first figure: O, N, M, L? Wait, no, O to L, L to M, M to N, N to O? Wait, no, the first figure has vertices O, N, M, L? Wait, O to N is 15 mi, N to M is 17 mi, M to L is 49 mi, L to O is 48 mi, and O to N to M to L to O? Wait, no, that's a quadrilateral? Wait, no, the angles: O (109°), N (116°), M (unknown), L (32°), and the sum of angles in a quadrilateral is 360°? Wait, no, wait the second figure: U, V, W, T, U? U to T is 48 mi, T to W is 49 mi, W to V is 17 mi, V to U? Wait, V to U is a side, and U to T is 48 mi. Wait, the angle at W is 103°, angle at U? Wait, the two figures are congruent, so corresponding angles are equal. Let's list the sides:

First figure sides: ON = 15 mi, NM = 17 mi, ML = 49 mi, LO = 48 mi.

Second figure sides: UV = 15 mi? Wait, no, V to W is 17 mi, W to T is 49 mi, T to U is 48 mi, U to V? Wait, U to V is 15 mi? Wait, the angle at W is 103°, angle at O is 109°, angle at N is 116°, angle at L is 32°, angle at M is? Wait, no, maybe it's a quadrilateral? Wait, no, the sum of angles in a quadrilateral is 360°. Wait, first figure angles: O (109°), N (116°), L (32°), M (x). Second figure angles: U (?), V (?), W (103°), T (?). Since they are congruent, corresponding angles are equal. Let's find the corresponding angle to M. The side NM is 17 mi, and in the second figure, VW is 17 mi (V to W is 17 mi). So angle at M corresponds to angle at W? Wait, no, angle at W is 103°, but wait, no, let's check the angles. Wait, in the first figure, angle at O is 109°, angle at N is 116°, angle at L is 32°, angle at M is x. In the second figure, angle at W is 103°, angle at U is?, angle at V is?, angle at T is?. Wait, sum of angles in a quadrilateral is 360°. Wait, first figure: 109 + 116 + 32 + x = 360? Wait, no, that's 4 angles, but maybe it's a pentagon? Wait, no, the first figure has 4 sides? Wait, O to N (15), N to M (17), M to L (49), L to O (48) – that's a quadrilateral (4 sides). So sum of angles in a quadrilateral is 360°. So 109 + 116 + 32 + x = 360? Wait, 109 + 116 = 225, 225 + 32 = 257, 360 - 257 = 103. Wait, but the second figure has an angle at W of 103°, so angle M corresponds to angle W? Wait, no, wait the sides: NM is 17 mi, VW is 17 mi. ML is 49 mi, WT is 49 mi. LO is 48 mi, TU is 48 mi. ON is 15 mi, UV is 15 mi. So the correspondence is O ↔ U, N ↔ V, M ↔ W, L ↔ T. So angle at M (first figure) corresponds to angle at W (second figure), which is 103°. Wait, but let's check the sum. Wait, first figure angles: O (109°), N (116°), L (32°), M (x). Second figure angles: U (109°), V (116°), T (32°), W (103°). So sum: 109 + 116 + 32 + 103 = 109+116=225, 225+32=257, 257+103=360, which matches the sum of angles in a quadrilateral (360°). So angle M corresponds to angle W, which is 103°. Wait, but wait, the first figure's angle at N is 116°, second figure's angle at V is 116°? Wait, no, the second figure's angle at W is 103°, so angle M is equal to angle W, which is 103°. Wait, but let's confirm the correspondence. The sides: ON = 15 mi (O to N), UV = 15 mi (U to V) – so O ↔ U, N ↔ V. NM = 17 mi (N to M), VW = 17 mi (V to W) – so M ↔ W. ML = 49 mi (M to L), WT = 49 mi (W to T) – so L ↔ T. LO = 48 mi (L to O), TU = 48 mi (T to U) – so O ↔ U, L ↔ T. So the angles: angle at O (109°) ↔ angle at U, angle at N (116°) ↔ angle at V, angle at M (x) ↔ angle at W (103°), angle at L (32°) ↔ angle at T. Therefore, m∠M = m∠W = 103°.

Wait, but let's check the sum of angles in a quadrilateral: 109 + 116 + 32 + 103 = 109+116=225, 225+32=257, 257+103=360. Correct, since sum of interior angles of a quadrilateral is 360°. So yes, m∠M is 103°.

Step3: Confirm Congruence

Since the figures are congruent, corresponding angles are equal. The side NM (17 mi) corresponds to VW (17 mi), so angle at M corresponds to angle at W, which is 103°. Thus, m∠M = 103°.

Answer:

103